FERMILAB-FN-0298

# Very Big Accelerators as Energy Producers

R.R. Wilson  Fermi National Accelerator Laboratory
Batavia, IL 60510 USA

(August 9, 1976)

One consequence of the application of superconductivity to accelerator
construction is that the power consumption of accelerators will become much
smaller. This raises the old possibility of using high energy protons to make
neutrons which are then absorbed by fertile uranium or thorium to make a
fissionable material like plutonium that can be burned in a nuclear reactor
111Shades of E. O. Lawrence’s MTA project! See Atomic Shield, page 425, by R.
Hewlett and F. Duncan. I have also learned that there is a project in Canada
to use an accelerator this way. See W. Metz, Science, July 23, 1976, Page
307.. The Energy Doubler/Saver being constructed at Fermilab is to be a
superconducting accelerator that will produce 1000 GeV protons. The expected
intensity of about $10^{12}$ protons per second corresponds to a beam power of
about 0.2 MW. The total power requirements of the Doubler will be about 20 MW
of which the injector complex will use approximately 13 MW, and the
refrigeration of the superconducting magnets will use about 7 MW. Thus the
beam power as projected is only a few orders of magnitude less than the accele
ator power. But each 1000 GeV proton will produce about 60,000 neutrons in
each nuclear cascade shower that is releaseq in a block of uranium, and then
most of these neutrons will be absorbed to produce 60,000 plutonium a toms.
Each of these when burned will Subsequently release about 0.2 GeV of fission
energy to make a total energy of 12,000 GeV (20 ergs) for each 1000 GeV
proton. Inasmuch as megawatts are involved, it appears to be worthwhile to
consider the cost of making the protons to see if there could be an overall
energy production.

A high energy proton accelerator seems at first glance to be an unlikely
device for producing energy in this manner but high energy does have a few
special advantages. One is that once the protons have been injected and
accepted by the accelerator at the low energy of injection, then the
subsequent losses during the acceleration to high energy are essentially zero.
Another is that space charge limitations of intensity become less important at
high energy. A third advantage is that the betatron oscillations of the beam
are damped as the energy increases. This means that the emittance of the beam
gets better as the energy grows and hence the beam can be transferred to rings
of smaller aperture (lower refrigeration costs) as the energy and hence ring
size increases. This assumes that the accelerator consists of a series of
magnet rings of ever increasing radius. Alternatively, more protons can be
injected into a given aperture and accelerated. The main point is that the
number of neutrons produced is roughly proportional to the beam power and this
can be made large by increasing both the intensity and the energy of the
protons. The intensity of an accelerator usually runs into a hard limit
imposed by space charge and resonance phenomenon, but the energy can be
increased without limit.

A disadvantage of very high proton energy is that the relative number of
neutrons produced in the nuclear shower gets somewhat less as the energy of
the initial proton increases because the fraction of the electromagnetic
component increases with energy. This adverse effect could be vitiated in part
by accelerating deuterons (or particles of even higher mass) so that the
energy per nucleon is lower. In any case, once a proton has gone through the
expensive and inefficient business of being produced and accelerated to about
10 GeV energy, then all the energy possible should be pumped into it during
the efficient part of the acceleration process that brings it to high energy.

Now let us consider the process a little more quantitatively. We assume that
the number of neutrons produced in a nuclear shower in U238 when $N$ protons
of energy $E$ are incident is proportional to the energy in the shower, i.e.,
to $aNE$, where the constant $a$ will have a value of roughly 60 neutrons per
GeV, about half of which come from fission of U238 Then the potential power
$P$ produced overall when these neutrons are nearly all absorbed to form
plutonium is given by

$P=0.2aNE-P_{0}-bNE,$

where 0.2 is the energy per fission in GeV, $P_{0}$ is the power required to
run the accelerator when no protons are accelerated, and $bNE$ is the power
used by the RF system to accelerate the protons. The constant $b$ is a measure
of the inefficiency of producing RF power from the electrical mains; it has a
value of roughly 2 although this might be improved to a value of about 1.5.

For a given accelerator, if $0.2a$ is greater than $b$, as it is for the
Energy Doubler, then there will be a proton intensity $N_{0}$ above which more
power will be available from the plutonium burned than will be used by the
accelerator. This value is given by

$N_{0}\approx P_{0}/10E.$

For the Energy Doubler at Fermilab plus all its injector stages, $P_{0}$ will
be roughly 20 MW, and $N_{0}$ then comes out to be about $2\times 10^{13}$
protons per second. This is about twenty times the expected intensity - but it
is far from being unattainable. An intensity of $10^{13}$ protons per second
will make about 15 Megawatts of fission energy available; this does not count
the energy put into the accelerator. For an overall production of 15 MW, an
intensity of $3\times 10^{13}$ protons would be required; or more generally

$P\approx 10(N-N_{0})E.$

Now the above calculations are pessimistic in that they assume that the beam
energy is thrown away. Furthermore there will be considerable fission energy
produced by fast neutrons being absorbed in the U238 during the course of the
cascade shower. Let us now assume that both of these forms of energy are
available for use. Mr. Andreas VanGinneken has made a rough computer
calculation following the shower development and assuming that each U238
fission leads to 3 Pu239 nuclei, but that the Pu239 fission is not used as a
source of neutrons. His results are tabulated in Table I and illustrate the
relative increase of the electromagnetic component with energy. Remaking the
calculations for the rougher break-even intensity made above, we find $N_{0}$
to be close to $10^{13}$ protons per second, an intensity which might be
attainable even in the Doubler by increasing the rate of pulsing, or by
resorting to a stacking process, or by adding a second ring of superconducting
magnets. He has also made the same calculation at 100 and 300 GeV.

It appears then that an accelerator that is similar to the Energy Doubler
could be made to be energy productive. Starting from the beginning, perhaps
the injector system could be made to use less ertergy and produce higher
intensities. One could also optimize the size of the rings so as to reduce the
refrigeration requirement and to raise the intensity capability. An important
improvement (and one assumed in all the above) would be to make at least two
magnet-rings back-to-back so that the electric energy could oscillate from one
to the other rather than just being put back into the power mains. This would
double the intensity and would make efficient use of the injection system, for
one ring could be loaded while the other was accelerating the protons
previously loaded into it. My guess is that the optimum energy of the
accelerator will be smaller, but this will depend on details of the injection
system and on the shape of the real estate that is available. Values of
$N_{0}$ for other energies are given in Table I: they vary roughly inversely
with the energy but 100 GeV gives fractionally about 30% more energy per
proton than does 1000 GeV.

Capital costs, of course, are equally significant. The bare-bones accelerator
might cost, as a very rough guess, about $200 million for a plant that would
produce the fuel to power a 100 MW fission plant. A larger installation might
of course cost relatively less per MW.

When the kind of intense proton beam considered above is to be absorbed on a
target, a serious problem arises. Even with our present intensities at 400
BeV, i.e., about $10^{12}$ protons/sec, targets tend to disappear. One can
imagine that the target might be a slurry of uranium oxide and heavy water.
Each pulse of protons will last about twenty microseconds, then the water will
turn to stearn and start to explode. This could move a piston in the classical
manner until the stearn has cooled (or has passed to a next stage) and has
been replaced again by the slurry. Such a dramatic device need not be used;
rather the stearn could be taken off in the more usual manner of any power
reactor and then used for the generation of energy.

If undepleted uranium were to be used (or were slightly enriched with Pu) then
a magnification of the neutrons would occur, but still without having a
supercritical reactor. This might bring the break-even intensity of protons
needed for power production down by another factor of ten, which would then
put it within range of the present Energy Doubler accelerator at Fermilab.

There are probably better ways of producing plutonium, but it does appear that
it would be feasible to construct an intense proton accelerator that would
produce more energy than it consumes. A further more careful study of the
accelerator as well as of the plutonium production in the cascade showers
would determine the optimum proton energy for such a device. One happy result
of all that intensity would be that a truly magnificent neutrino source could
be produced!

I wish to thank Mr. A. Van Ginneken for making the nuclear shower calculation.

Table I
---
Calculation of A. VanGinneken for protons on
a large U238 beam dump
| 100 Gev | 300 Gev | 1000 Gev
Ioniz. of Hadron Shower* | .34 | .29 | .23
E. M. Showers* | .32 | .40 | .48
U238 Fission* | 2.2 | 1.9 | 1.6
Pu239 Fission* | 13.8 | 11.8 | 10.2
Total* | 16.6 | 14.3 | 12.5
$N^{0}$ | $8.6\times 10^{13}$ | $3.4\times 10^{13}$ | $1.2\times 10^{13}$
(Pu nuclei)/proton | 7000 | 18,000 | 62,000
Total Energy/1013 protons | 2.7 | 6.9 | 20 MW

* Fraction of incident energy recovered: calculations assume that each U238 fission makes three Pu239 nuclei. The fission of Pu239 has not been considered as a source of neutrons.


#  Non-density of $C^{0}$-stable mappings
on non-compact manifolds

Shunsuke Ichiki  Department of Mathematical and Computing Science, School of
Computing, Tokyo Institute of Technology, Tokyo 152-8552, Japan
<EMAIL_ADDRESS>
###### Abstract.

In 1973, Mather showed that the set of proper $C^{0}$-stable mappings is dense
in the set of all proper mappings, which implies that the set of
$C^{0}$-stable mappings is dense in the set of all mappings if the source
manifold is compact. In this paper, we show that the set of $C^{0}$-stable
mappings is never dense in the set of all mappings if the source manifold is
non-compact. To prove it, we provide a more essential result by using the
notion of topologically critical points.

###### Key words and phrases:

$C^{0}$-stable mapping, Whitney $C^{\infty}$-topology, topologically critical
point

###### 2020 Mathematics Subject Classification:

57R45, 58K30

##  1\. Introduction

Let $N$ and $P$ be $C^{\infty}$-manifolds. Two $C^{\infty}$-mappings $f$ and
$g$ from $N$ into $P$ are said to be $C^{\infty}$-equivalent (resp.
$C^{0}$-equivalent) if there exist $C^{\infty}$-diffeomorphisms (resp.
homeomorphisms) $\Phi:N\to N$ and $\Psi:P\to P$ such that $g=\Psi\circ
f\circ\Phi^{-1}$. We denote the space of all $C^{\infty}$-mappings of $N$ into
$P$ (resp. the space of all proper $C^{\infty}$-mappings of $N$ into $P$)
endowed with the Whitney $C^{\infty}$-topology by $C^{\infty}(N,P)$ (resp.
$C^{\infty}_{pr}(N,P)$). We say that $f$ is _$C^{\infty}$ -stable_ (resp.
_$C^{0}$ -stable_) if there exists an open neighborhood $\mathscr{U}$ of $f$
such that any mapping in $\mathscr{U}$ is $C^{\infty}$-equivalent (resp.
$C^{0}$-equivalent) to $f$. In what follows, unless otherwise stated, all
manifolds and mappings are of class $C^{\infty}$.

On the $C^{\infty}$-stability, in a celebrated series around 1970 [5, 6, 7, 8,
9, 10], Mather established a significant theory and he gave a characterization
of the density of proper $C^{\infty}$-stable mappings in
$C^{\infty}_{pr}(N,P)$ as follows:

######  Theorem 1.1 ([10]).

Let $N$ and $P$ be manifolds of dimensions $n$ and $p$, respectively. Then,
the set of all proper $C^{\infty}$-stable mappings is dense in
$C^{\infty}_{pr}(N,P)$ if and only if the pair $(n,p)$ satisfies one of the
following conditions.

  1. $(1)$

$n<\frac{6}{7}p+\frac{8}{7}$ and $p-n\geq 4$

  2. $(2)$

$n<\frac{6}{7}p+\frac{9}{7}$ and $3\geq p-n\geq 0$

  3. $(3)$

$p<8$ and $p-n=-1$

  4. $(4)$

$p<6$ and $p-n=-2$

  5. $(5)$

$p<7$ and $p-n\leq-3$

A dimension pair $(n,p)$ is called _nice_ if it satisfies one of the
conditions (1)–(5) in Theorem 1.1. Since
$C^{\infty}_{pr}(N,P)=C^{\infty}(N,P)$ in the case where $N$ is compact,
Theorem 1.1 yields a characterization of the density of $C^{\infty}$-stable
mappings in $C^{\infty}(N,P)$ in the case.

After that, the case where a source manifold is non-compact was considered by
Dimca, and in 1979, he gave the following result.

######  Proposition 1.2 ([1]).

Let $N$ be a non-compact manifold. Then, the set of all $C^{\infty}$-stable
mappings is not dense in $C^{\infty}(N,\mathbb{R})$.

Then, in [4], for a non-compact source manifold $N$ and an arbitrary target
manifold $P$, the following has been shown rigorously:

######  Theorem 1.3 ([4]).

Let $N$ be a non-compact manifold, and $P$ a manifold. Then, the set of all
$C^{\infty}$-stable mappings is never dense in $C^{\infty}(N,P)$.

By combining Mather’s theorem (Theorem 1.1) and Theorem 1.3, we obtain the
following characterization of the density of $C^{\infty}$-stable mappings in
$C^{\infty}(N,P)$ in the case where $N$ is not necessarily compact.

######  Corollary 1.4 ([10, 4]).

Let $N$ and $P$ be manifolds of dimensions $n$ and $p$, respectively. Then,
the set of all $C^{\infty}$-stable mappings is dense in $C^{\infty}(N,P)$ if
and only if $N$ is compact and $(n,p)$ is nice.

On the $C^{0}$-stability, Mather established the following theorem in 1973,
which is so significant as well as Theorem 1.1, and the bibles [2, 3, 12] have
been published.

######  Theorem 1.5 ([11]).

Let $N$ and $P$ be manifolds. Then, the set of all proper $C^{0}$-stable
mappings is dense in $C^{\infty}_{pr}(N,P)$.

Theorem 1.5 implies that the set of all $C^{0}$-stable mappings is always
dense in $C^{\infty}(N,P)$ if $N$ is compact. In this paper, we give the
following result in the case where $N$ is non-compact.

######  Theorem 1.6.

Let $N$ be a non-compact manifold, and $P$ a manifold. Then, the set of all
$C^{0}$-stable mappings is never dense in $C^{\infty}(N,P)$.

In Section 2, we state a more essential result (see Theorem 2.2) than Theorem
1.6 by using the notion of topologically critical points, which is the main
theorem of this paper, and Theorem 1.6 follows from the result as a corollary.
We note that Theorem 1.6 implies Theorem 1.3 since any $C^{\infty}$-stable
mapping is $C^{0}$-stable. By combining Mather’s theorem on $C^{0}$-stability
(Theorem 1.5) and Theorem 1.6, we also obtain a characterization of the
density of $C^{0}$-stable mappings in $C^{\infty}(N,P)$ in the case where $N$
is not necessarily compact as follows:

######  Corollary 1.7.

Let $N$ and $P$ be manifolds. Then, the set of all $C^{0}$-stable mappings is
dense in $C^{\infty}(N,P)$ if and only if $N$ is compact.

The remainder of this paper is organized as follows. In Section 2, we state
the main theorem, which is a more essential result than Theorem 1.6, and
preparations for the proof. In Section 3, we show the main theorem.

##  2\. Main theorem and preparations for the proof

Let $N$ and $P$ be manifolds, and $f:N\to P$ a mapping. A point $q\in N$ is
called a _topologically critical point_ (resp. _critical point_) of $f$ if
$f|_{U}:U\to P$ is not an open mapping for any open neighborhood $U$ of $q$
(resp. $\operatorname{rank}df_{q}<\dim P$). We give the following remark on
topologically critical points.

######  Remark 2.1.

  1. (1)

If $q\in N$ is a topologically critical point of $f:N\to P$, then $q$ is a
critical point as follows: Suppose that $q\in N$ is not a critical point.
Then, since $\operatorname{rank}df_{q}\geq\dim P$, there exists an open
neighborhood $U$ such that $f|_{U}:U\to P$ is an open mapping by the implicit
function theorem, which contradicts the hypothesis that $q$ is a topologically
critical point.

  2. (2)

A critical point is not necessarily a topologically critical point. For
example, let $f:\mathbb{R}\to\mathbb{R}$ be the function defined by
$f(x)=x^{3}$. Then, $x=0$ is a critical point of $f$, however the point is not
a topologically critical point.

  3. (3)

Suppose that $f,g\in C^{\infty}(N,P)$ are $C^{0}$-equivalent, that is,
$g=\Psi\circ f\circ\Phi^{-1}$, where $\Phi:N\to N$ and $\Psi:P\to P$ are
homeomorphisms. If $q\in N$ is a topologically critical point of $f$, then
$\Phi(q)$ is a topologically critical point of $g$. Namely, topologically
critical points can be preserved by homeomorphisms although usual critical
points are not necessarily preserved by them.

The following is the main theorem, and Theorem 1.6 is an easy consequence of
this theorem.

######  Theorem 2.2.

Let $N$ be a non-compact manifold, and $P$ a manifold. Then, there exists a
non-empty open subset $\mathscr{U}$ of $C^{\infty}(N,P)$ such that for any
positive integer $d$ satisfying $d\geq 2$,

$\displaystyle\mathscr{U}_{d}=\set{f\in\mathscr{U}}{\mbox{$f$ has $d$
topologically critical points with the same image}}$

is dense in $\mathscr{U}$. In particular, any mapping in $\mathscr{U}$ is not
$C^{0}$-stable.

In Theorem 2.2, when we show that any mapping in $\mathscr{U}$ is not
$C^{0}$-stable, we use the simple fact that topologically critical points are
preserved by homeomorphisms as mentioned in Remark 2.13, which is an advantage
of topologically critical points compared to usual critical points.

In what follows, for a given positive integer $m$, we denote the origin
$(0,\ldots,0)$ of $\mathbb{R}^{m}$ by $0$, the Euclidean norm of
$x\in\mathbb{R}^{m}$ by $\|x\|$, and the $m$-dimensional open ball with center
$x\in\mathbb{R}^{m}$ and radius $r>0$ by $B^{m}(x,r)$, that is,

$\displaystyle
B^{m}(x,r)=\set{x^{\prime}\in\mathbb{R}^{m}}{\|x-x^{\prime}\|<r}.$

For a set (resp. a topological space) $X$ and a subset $A$ of $X$, we denote
the complement of $A$ (resp. the closure of $A$) by $A^{c}$ (resp.
$\overline{A}$).

The former assertion on a usual critical point of the following lemma is the
same as [4, Lemma 2.1], and we update the lemma by adding the latter assertion
on topologically critical points as follows:

######  Lemma 2.3.

Let $f=(f_{1},\ldots,f_{p}):B^{n}(0,r)\to\mathbb{R}^{p}$ $(r>0)$ be a mapping
such that

$\displaystyle f_{p}(x)=\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}+a,$

where $a$ is a real number and $x=(x_{1},\ldots,x_{n})$. If
$g=(g_{1},\ldots,g_{p}):B^{n}(0,r)\to\mathbb{R}^{p}$ satisfies

(2.1) $\displaystyle\sqrt{\sum_{i=1}^{n}\left(\frac{\partial f_{p}}{\partial
x_{i}}(x)-\frac{\partial g_{p}}{\partial x_{i}}(x)\right)^{2}}<\frac{r}{2}$

for any $x\in B^{n}(0,r)$, then there exists a critical point $x_{0}$ of $g$
in $B^{n}(0,r)$. Moreover, if $\mathrm{Hess}(g_{p})_{x_{0}}$ is positive
definite, then $x_{0}$ is a topologically critical point of $g$, where
$\mathrm{Hess}(g_{p})_{x_{0}}$ is the Hessian matrix of $g_{p}$ at $x_{0}$.

###### Proof of Lemma 2.3.

By the same proof as [4, Lemma 2.1], it follows that there exists a critical
point $x_{0}$ of $g$ in $B^{n}(0,r)$ such that $\frac{\partial g_{p}}{\partial
x_{i}}(x_{0})=0$ for any $i\in\set{1,\ldots,n}$.

Suppose that $\mathrm{Hess}(g_{p})_{x_{0}}$ is positive definite. Since
$x_{0}$ is also a critical point of $g_{p}$, there exists a coordinate
neighborhood $(U,\varphi)$ of $B^{n}(0,r)$ containing $x_{0}$
($\varphi(x_{0})=0\in\mathbb{R}^{n}$) such that
$g_{p}\circ\varphi^{-1}:\varphi(U)\to\mathbb{R}$ has the following form:

$\displaystyle(g_{p}\circ\varphi^{-1})(t_{1},\ldots,t_{n})=g_{p}(x_{0})+t_{1}^{2}+\cdots+t_{n}^{2},$

which implies that $x_{0}$ is a topologically critical point of $g$. ∎

##  3\. Proof of the main theorem

The proof is divided into the following three steps: In STEP 1, we construct
the $C^{\infty}$ mapping $f:N\to P$ defined by (3.4). In STEP 2, we construct
the open neighborhood $\mathscr{U}$ of $f$ in $C^{\infty}(N,P)$ given by
(3.6), and we provide a lemma on properties of a mapping in $\mathscr{U}$ (see
Lemma 3.1). Finally, in STEP 3, after preparing two lemmas (Lemmas 3.2 and
3.3), we show that $\mathscr{U}_{d}$ is dense in $\mathscr{U}$, and by using
this assertion and the simple fact that topologically critical points are
preserved by homeomorphisms, we prove that any mapping in $\mathscr{U}$ is not
$C^{0}$-stable.

The method of the proof is almost the same as that of Theorem 1.3, but the
most important difference is the addition of condition (c) in the definition
of $O_{\alpha}$ in 3.5 of STEP 2 to deal with topological critical points
instead of ordinary critical points (more precisely, to use Lemma 2.3). In
fact, in the proof of Theorem 1.3, there are only conditions (a) and (b) in
the definition of $O_{\alpha}$, and the open set $\mathscr{U}$ in 3.6 is
defined by using an open subset of the 1-jet space $J^{1}(N,P)$. On the other
hand, in this proof, since we add condition (c), we define $\mathscr{U}$ in
3.6 using an open set in the 2-jet space $J^{2}(N,P)$.

STEP 1 is the same as that in Theorem 1.3, but is also described in this paper
for the sake of readers’ convenience since in this first step we introduce
some symbols that will be subsequently used.

STEP 1. Set $n=\dim N$, $p=\dim P$ and $\ell=2n+1$. By Whitney’s embedding
theorem, there exist an embedding $F:N\to\mathbb{R}^{\ell}$ such that $F(N)$
is a closed subset of $\mathbb{R}^{\ell}$. Then, there exists a point
$z_{0}\in\mathbb{R}^{\ell}\setminus F(N)$. Since $N$ is non-compact, $F(N)$ is
also non-compact. Thus, $F(N)$ is not bounded, which implies that there exists
a sequence $\set{R_{\alpha}}_{\alpha\in\mathbb{N}}$ of positive real numbers
and a sequence $\set{z_{\alpha}}_{\alpha\in\mathbb{N}}$ of points in
$\mathbb{R}^{\ell}$ such that

  * •

$R_{\alpha}<R_{\alpha+1}$ for any $\alpha\in\mathbb{N}$ and
$\displaystyle\lim_{\alpha\to\infty}R_{\alpha}=\infty$,

  * •

$z_{\alpha}\in
F(N)\cap(B^{\ell}(z_{0},R_{\alpha+1})\setminus\overline{B^{\ell}(z_{0},R_{\alpha})})$
for any $\alpha\in\mathbb{N}$.

Let $\alpha$ be a positive integer. Set $q_{\alpha}=F^{-1}(z_{\alpha})$. Here,
note that

$\displaystyle
F^{-1}(B^{\ell}(z_{0},R_{\alpha+1})\setminus\overline{B^{\ell}(z_{0},R_{\alpha})})$

is an open neighborhood of $q_{\alpha}$. Then, there exists a coordinate
neighborhood $(U_{\alpha},\varphi_{\alpha})$ of $N$ with the following
properties:

  * •

$\overline{U_{\alpha}}$ is compact,

  * •

$q_{\alpha}\in U_{\alpha}\subset
F^{-1}(B^{\ell}(z_{0},R_{\alpha+1})\setminus\overline{B^{\ell}(z_{0},R_{\alpha})})$,

  * •

$\varphi_{\alpha}(q_{\alpha})=0\in\mathbb{R}^{n}$.

Moreover, there exist an open neighborhood $U_{\alpha}^{\prime}$ of
$q_{\alpha}$ and $\rho_{\alpha}:N\to\mathbb{R}$ ($0\leq\rho_{\alpha}(q)\leq
1$) such that

  * •

$\overline{U_{\alpha}^{\prime}}\subset U_{\alpha}$,

  * •

$\rho_{\alpha}(q)=1$ for any $q\in\overline{U_{\alpha}^{\prime}}$,

  * •

$\operatorname{supp}\rho_{\alpha}\subset U_{\alpha}$,

where $\operatorname{supp}\rho_{\alpha}=\overline{\set{q\in
N}{\rho_{\alpha}(q)\not=0}}$. Notice that $\operatorname{supp}\rho_{\alpha}$
is compact since $\overline{U_{\alpha}}$ is compact. By choosing
$U_{\alpha}^{\prime}$ smaller for each $\alpha\in\mathbb{N}$ we can assume
that

  * •

$\varphi_{\alpha}(U_{\alpha}^{\prime})=B^{n}(0,r_{\alpha})$,

  * •

$\displaystyle\lim_{\alpha\to\infty}r_{\alpha}=0$,

where each $r_{\alpha}$ is a positive real number.

Let $\gamma=(\gamma_{1},\ldots,\gamma_{p}):\mathbb{N}\to\mathbb{Q}^{p}$ be a
bijection, and let
$\eta_{\alpha}:\varphi_{\alpha}(U_{\alpha})\to\mathbb{R}^{p}$ be the mapping
defined by

$\displaystyle\eta_{\alpha}(x)=\displaystyle\left(\gamma_{1}(\alpha),\ldots,\gamma_{p-1}(\alpha),\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}+\gamma_{p}(\alpha)\right)$

for each $\alpha\in\mathbb{N}$, where $x=(x_{1},\ldots,x_{n})$. Let $(V,\psi)$
be a coordinate neighborhood of $P$ that satisfies $\psi(V)=\mathbb{R}^{p}$.
Since $U_{\alpha}\cap U_{\beta}=\varnothing$ if $\alpha\not=\beta$, we can
define $f:N\to P$ as follows:

(3.4) $\displaystyle
f(q)=\left\\{\begin{array}[]{ll}\displaystyle\psi^{-1}(\rho_{\alpha}(q)(\eta_{\alpha}\circ\varphi_{\alpha})(q))&\mbox{if
$q\in U_{\alpha}$},\\\ \\\ \displaystyle\psi^{-1}(0)&\mbox{if
$q\not\in\bigcup_{\alpha\in\mathbb{N}}U_{\alpha}$}.\end{array}\right.$

We show that $f$ is of class $C^{\infty}$. Let $q\in N$ be any point. If
$q\in\bigcup_{\alpha\in\mathbb{N}}U_{\alpha}$, then by the definition of $f$
it is clear that $f$ is of class $C^{\infty}$ at $q$. Thus, we consider the
case $q\in(\bigcup_{\alpha\in\mathbb{N}}U_{\alpha})^{c}$. Since
$\displaystyle\lim_{\alpha\to\infty}R_{\alpha}=\infty$, there exists
$\beta\in\mathbb{N}$ such that $q\in F^{-1}(B^{\ell}(z_{0},R_{\beta}))$. For
simplicity, set

$\displaystyle
A=F^{-1}(B^{\ell}(z_{0},R_{\beta}))\cap\left(\bigcup_{\alpha\in\mathbb{N}}\operatorname{supp}\rho_{\alpha}\right)^{c}.$

Note that $q\in A$. Since $R_{\alpha}<R_{\alpha+1}$ for any
$\alpha\in\mathbb{N}$, we have

$\displaystyle
F^{-1}(B^{\ell}(z_{0},R_{\beta}))\subset(\operatorname{supp}\rho_{\alpha})^{c}$

for each $\alpha\in\mathbb{N}$ satisfying $\alpha>\beta$. Thus, we obtain

$\displaystyle A$
$\displaystyle=F^{-1}(B^{\ell}(z_{0},R_{\beta}))\cap\left(\bigcap_{\alpha\in\mathbb{N}}(\operatorname{supp}\rho_{\alpha})^{c}\right)$
$\displaystyle=F^{-1}(B^{\ell}(z_{0},R_{\beta}))\cap\left(\bigcap_{\alpha\leq\beta}(\operatorname{supp}\rho_{\alpha})^{c}\right),$

which implies that $A$ is an open subset of $N$. Since $\rho_{\alpha}|_{A}$ is
a constant function with a constant value $0$ for each $\alpha\in\mathbb{N}$,
the mapping $f|_{A}$ is also constant. Thus, $f$ is of class $C^{\infty}$ at
$q$.

STEP 2. In this step, we construct the open neighborhood $\mathscr{U}$ of $f$
in $C^{\infty}(N,P)$ given by (3.6), and we provide a lemma on properties of a
mapping in $\mathscr{U}$. Since $z_{0}\in\mathbb{R}^{\ell}\setminus F(N)$, we
can define the following continuous function $\delta:N\to\mathbb{R}$:

$\displaystyle\delta(q)=\frac{1}{\left\|F(q)-z_{0}\right\|}.$

Let $\pi:J^{2}(N,P)\to N\times P$ be the natural projection defined by
$\pi(j^{2}g(q))=(q,g(q))$. Then, for each $\alpha\in\mathbb{N}$, set

(3.5) $\displaystyle O_{\alpha}=\set{j^{2}g(q)\in\pi^{-1}(U_{\alpha}\times
V)}{\mbox{$j^{2}g(q)$ satisfies \lx@cref{creftype~refnum}{eq:t},
\lx@cref{creftype~refnum}{eq:d} and \lx@cref{creftype~refnum}{eq:hess}}},$

where

  1. (a)

$\left\|(\psi\circ f)(q)-(\psi\circ g)(q)\right\|<\delta(q)$,

  2. (b)

$\displaystyle\sqrt{\sum_{i=1}^{n}\left(\frac{\partial(\psi_{p}\circ
f\circ\varphi_{\alpha}^{-1})}{\partial
x_{i}}(\varphi_{\alpha}(q))-\frac{\partial(\psi_{p}\circ
g\circ\varphi_{\alpha}^{-1})}{\partial
x_{i}}(\varphi_{\alpha}(q))\right)^{2}}<\frac{r_{\alpha}}{2}$,

  3. (c)

$\mathrm{Hess}(\psi_{p}\circ
g\circ\varphi_{\alpha}^{-1})_{\varphi_{\alpha}(q)}$ is positive definite,

where $\psi_{p}$ is the $p$-th component of $\psi$. From a, b and c, it is not
hard to see that $O_{\alpha}$ is an open subset of $J^{2}(N,P)$.

We prove that
$\bigcap_{\alpha\in\mathbb{N}}(\overline{U_{\alpha}^{\prime}})^{c}$ is an open
subset of $N$. Let
$q\in\bigcap_{\alpha\in\mathbb{N}}(\overline{U_{\alpha}^{\prime}})^{c}$ be any
point. Since $\displaystyle\lim_{\alpha\to\infty}R_{\alpha}=\infty$, there
exists $\beta\in\mathbb{N}$ such that $q\in
F^{-1}(B^{\ell}(z_{0},R_{\beta}))$. Since $R_{\alpha}<R_{\alpha+1}$ for each
$\alpha\in\mathbb{N}$, we obtain
$F^{-1}(B^{\ell}(z_{0},R_{\beta}))\subset(\overline{U_{\alpha}^{\prime}})^{c}$
for any $\alpha\in\mathbb{N}$ satisfying $\alpha>\beta$, which implies that

$\displaystyle
F^{-1}(B^{\ell}(z_{0},R_{\beta}))\cap\left(\bigcap_{\alpha\leq\beta}(\overline{U_{\alpha}^{\prime}})^{c}\right)\subset\bigcap_{\alpha\in\mathbb{N}}(\overline{U_{\alpha}^{\prime}})^{c}.$

Since the left-hand side of the above expression is an open neighborhood of
$q$, it follows that
$\bigcap_{\alpha\in\mathbb{N}}(\overline{U_{\alpha}^{\prime}})^{c}$ is open.
Thus, since $\pi$ is continuous,

$\displaystyle
O:=\left(\bigcup_{\alpha\in\mathbb{N}}O_{\alpha}\right)\cup\pi^{-1}\left(\left(\bigcap_{\alpha\in\mathbb{N}}\left(\overline{U_{\alpha}^{\prime}}\right)^{c}\right)\times
V\right)$

is open in $J^{2}(N,P)$. Therefore, we can construct the following open subset
of $C^{\infty}(N,P)$:

(3.6) $\displaystyle\mathscr{U}:=\set{g\in C^{\infty}(N,P)}{j^{2}g(N)\subset
O}.$

By showing that $j^{2}f(N)\subset O$, we will prove that
$\mathscr{U}\not=\varnothing$. Let $j^{2}f(q)$ $(q\in N)$ be any element of
$j^{2}f(N)$. If there exists $\alpha\in\mathbb{N}$ such that $q\in
U_{\alpha}$, then $j^{2}f(q)\in O_{\alpha}$ $(\subset O)$ since $f(q)\in V$
and $j^{2}f(q)$ clearly satisfies a, b and c. When
$q\not\in\bigcup_{\alpha\in\mathbb{N}}U_{\alpha}$, since

(3.7) $\displaystyle
N=\left(\bigcup_{\alpha\in\mathbb{N}}U_{\alpha}\right)\cup\left(\bigcap_{\alpha\in\mathbb{N}}\left(\overline{U_{\alpha}^{\prime}}\right)^{c}\right),$

it must follow that
$q\in\bigcap_{\alpha\in\mathbb{N}}\left(\overline{U_{\alpha}^{\prime}}\right)^{c}$.
Therefore, since $f(q)\in V$, we obtain

$\displaystyle
j^{2}f(q)\in\pi^{-1}\left(\left(\bigcap_{\alpha\in\mathbb{N}}(\overline{U_{\alpha}^{\prime}})^{c}\right)\times
V\right)\ (\subset O),$

which implies that $\mathscr{U}\not=\varnothing$.

The following lemma describes properties of a mapping in $\mathscr{U}$ and it
is an upgrade of [4, Lemma 3.1] to a claim about a topological critical point
instead of a usual critical point.

######  Lemma 3.1.

For any mapping $g\in\mathscr{U}$, we have $g(N)\subset V$ and there exists a
sequence $\set{q_{\alpha}^{\prime}}_{\alpha\in\mathbb{N}}$ of points in $N$
with the following properties.

  1. $(1)$

For each $\alpha\in\mathbb{N}$, $q_{\alpha}^{\prime}$ is a topologically
critical point of $g$ in $U_{\alpha}^{\prime}$.

  2. $(2)$

The set $\set{g(q_{\alpha}^{\prime})}{\alpha\in\mathbb{N}}$ is dense in $V$.

###### Proof of Lemma 3.1.

From the definition of $\mathscr{U}$, we have $g(N)\subset V$.

Let $\alpha$ be any positive integer. Then, we have

$\displaystyle(\psi_{p}\circ
f\circ\varphi_{\alpha}^{-1})(x)=\frac{1}{2}\sum_{i=1}^{n}x_{i}^{2}+\gamma_{p}(\alpha)$

for any $x=(x_{1},\ldots,x_{n})\in\varphi_{\alpha}(U_{\alpha}^{\prime})$
$(=B^{n}(0,r_{\alpha}))$. For any $q\in U_{\alpha}^{\prime}$, we obtain
$j^{2}g(q)\in O_{\alpha}$ since we have 3.7 and $U_{\alpha}^{\prime}$ is
contained in $U_{\alpha}$ which does not intersect with $U_{\beta}$
$(\beta\not=\alpha)$. Hence, $(\psi_{p}\circ
g\circ\varphi_{\alpha}^{-1})|_{B^{n}(0,r_{\alpha})}$ satisfies b and c, which
implies that there exists a topologically critical point of $(\psi_{p}\circ
g\circ\varphi_{\alpha}^{-1})|_{B^{n}(0,r_{\alpha})}$ in $B^{n}(0,r_{\alpha})$
by Lemma 2.3. Namely, there exists a topologically critical point of $g$ in
$U_{\alpha}^{\prime}$. We denote this point by $q_{\alpha}^{\prime}$.

Since $\set{q_{\alpha}^{\prime}}_{\alpha\in\mathbb{N}}$ satisfies 1, it is
sufficient to prove that the sequence of points also satisfies 2. Let
$V^{\prime}$ be any open subset of $V$. We show that
$\set{g(q_{\alpha}^{\prime})}{\alpha\in\mathbb{N}}\cap
V^{\prime}\not=\varnothing$. Then, by choosing $V^{\prime}$ smaller, we can
assume that $\psi(V^{\prime})=B^{p}(y_{0},\varepsilon)$, where $y_{0}$ is a
point of $\mathbb{R}^{p}$ and $\varepsilon$ is a positive real number. Note
that for each $\alpha\in\mathbb{N}$, we have

(3.8) $\begin{split}\left\|(\psi\circ
g)(q_{\alpha}^{\prime})-y_{0}\right\|\leq&\left\|(\psi\circ
g)(q_{\alpha}^{\prime})-(\psi\circ f)(q_{\alpha}^{\prime})\right\|+\\\
&\left\|(\psi\circ f)(q_{\alpha}^{\prime})-(\psi\circ
f)(q_{\alpha})\right\|+\left\|(\psi\circ
f)(q_{\alpha})-y_{0}\right\|.\end{split}$

Since

$\displaystyle\delta(q_{\alpha}^{\prime})=\frac{1}{\left\|F(q_{\alpha}^{\prime})-z_{0}\right\|}<\frac{1}{R_{\alpha}}$

for any $\alpha\in\mathbb{N}$ and
$\displaystyle\lim_{\alpha\to\infty}R_{\alpha}=\infty$, there exists
$\alpha_{1}\in\mathbb{N}$ such that
$\delta(q_{\alpha}^{\prime})<\frac{\varepsilon}{3}$ for any
$\alpha\in\mathbb{N}$ satisfying $\alpha\geq\alpha_{1}$. Here, note that for
each $\alpha\in\mathbb{N}$, we get

$\displaystyle\left\|(\psi\circ g)(q_{\alpha}^{\prime})-(\psi\circ
f)(q_{\alpha}^{\prime})\right\|<\delta(q_{\alpha}^{\prime})$

by a since $j^{2}g(q_{\alpha}^{\prime})\in O_{\alpha}$. Thus, it follows that
for any $\alpha\in\mathbb{N}$,

(3.9) $\displaystyle\alpha\geq\alpha_{1}\Longrightarrow\left\|(\psi\circ
g)(q_{\alpha}^{\prime})-(\psi\circ
f)(q_{\alpha}^{\prime})\right\|<\frac{\varepsilon}{3}.$

For each $\alpha\in\mathbb{N}$, since $q_{\alpha},q_{\alpha}^{\prime}\in
U_{\alpha}^{\prime}$, we have

$\displaystyle\left\|(\psi\circ f)(q_{\alpha}^{\prime})-(\psi\circ
f)(q_{\alpha})\right\|=\left\|\eta_{\alpha}(\varphi_{\alpha}(q_{\alpha}^{\prime}))-\gamma(\alpha)\right\|=\frac{\left\|\varphi_{\alpha}(q_{\alpha}^{\prime})\right\|^{2}}{2}<\frac{r_{\alpha}^{2}}{2}.$

Since $\displaystyle\lim_{\alpha\to\infty}r_{\alpha}=0$, there exists
$\alpha_{2}\in\mathbb{N}$ such that for any $\alpha\in\mathbb{N}$,

(3.10) $\displaystyle\alpha\geq\alpha_{2}\Longrightarrow\left\|(\psi\circ
f)(q_{\alpha}^{\prime})-(\psi\circ
f)(q_{\alpha})\right\|<\frac{\varepsilon}{3}.$

Since $(\psi\circ f)(q_{\alpha})=\gamma(\alpha)$ for each
$\alpha\in\mathbb{N}$, we have

$\displaystyle\set{(\psi\circ
f)(q_{\alpha})}{\alpha\in\mathbb{N}}=\mathbb{Q}^{p}.$

Hence, there exists $\alpha_{3}\in\mathbb{N}$ such that
$\alpha_{3}>\max\set{\alpha_{1},\alpha_{2}}$ and

(3.11) $\displaystyle\left\|(\psi\circ
f)(q_{\alpha_{3}})-y_{0}\right\|<\frac{\varepsilon}{3}.$

Thus, we get $\left\|(\psi\circ
g)(q_{\alpha_{3}}^{\prime})-y_{0}\right\|<\varepsilon$ by (3.8) to 3.11, which
implies that $g(q_{\alpha_{3}}^{\prime})\in V^{\prime}$. ∎

STEP 3. The purpose of this step is to show that $\mathscr{U}_{d}$ is dense in
$\mathscr{U}$, where $d\geq 2$ is a given integer. Let $g\in\mathscr{U}$ be an
arbitrary mapping, and let $\mathscr{U}_{g}$ be any open neighborhood of $g$.
Then, there exist a non-negative integer $k$ and an open set $O^{\prime}$ of
$J^{k}(N,P)$ such that

$\displaystyle g\in\set{h\in C^{\infty}(N,P)}{j^{k}h(N)\subset
O^{\prime}}\subset\mathscr{U}_{g}.$

For the proof, it is sufficient to show that there exists a mapping
$h\in\mathscr{U}_{d}$ such that $j^{k}h(N)\subset O^{\prime}$.

For any $\alpha\in\mathbb{N}$ and $c\in\mathbb{R}^{p}$, let $G_{\alpha,c}:N\to
P$ be the mapping defined by

$\displaystyle G_{\alpha,c}=\psi^{-1}\circ(\psi\circ g+\rho_{\alpha}c).$

######  Lemma 3.2 ([4, Lemma 3.2]).

Let $\alpha$ be any positive integer. Then, there exists a positive real
number $r_{\alpha}^{\prime}$ such that $j^{k}G_{\alpha,c}(N)\subset
O^{\prime}$ for any $c\in B^{p}(0,r_{\alpha}^{\prime})$.

Since $g\in\mathscr{U}$, note that there exists a sequence
$\set{q_{\alpha}^{\prime}}_{\alpha\in\mathbb{N}}$ of points in $N$ satisfying
1 and 2 of Lemma 3.1. The following lemma can be also shown by the same method
as [4, Lemma 3.3].

######  Lemma 3.3.

Let $m$ be any positive integer. Then, there exist $(m+1)$ distinct positive
integers $\alpha_{1},\ldots,\alpha_{m+1}$ and $m$ positive real numbers
$r_{\alpha_{1}}^{\prime},\ldots,r_{\alpha_{m}}^{\prime}$
$(r_{\alpha_{1}}^{\prime}>\cdots>r_{\alpha_{m}}^{\prime})$ such that for any
$j\in\set{1,\ldots,m}$,

  1. $(1)$

$j^{k}G_{\alpha_{j},c}(N)\subset O^{\prime}$ for any $c\in
B^{p}(0,r_{\alpha_{j}}^{\prime})$,

  2. $(2)$

$\left\|(\psi\circ g)(q_{\alpha_{j+1}}^{\prime})-(\psi\circ
g)(q_{\alpha_{j}}^{\prime})\right\|<\displaystyle\frac{r_{\alpha_{j}}^{\prime}}{d-1}$,
where $d\geq 2$ is given in Theorem 2.2.

###### Proof of Lemma 3.3.

We will prove this lemma by induction on $m$.

Let $\alpha_{1}$ be a positive integer. By Lemma 3.2, there exists a positive
real number $r_{\alpha_{1}}^{\prime}$ such that
$j^{k}G_{\alpha_{1},c}(N)\subset O^{\prime}$ for any $c\in
B^{p}(0,r_{\alpha_{1}}^{\prime})$. From Lemma 3.1 2, there exists
$\alpha_{2}\in\mathbb{N}\setminus\set{\alpha_{1}}$ satisfying

$\displaystyle\left\|(\psi\circ g)(q_{\alpha_{2}}^{\prime})-(\psi\circ
g)(q_{\alpha_{1}}^{\prime})\right\|<\frac{r_{\alpha_{1}}^{\prime}}{d-1}.$

Thus, the case $m=1$ holds.

We assume that the lemma holds for $m=i$, where $i$ is a positive integer. By
Lemma 3.2, there exists a positive real number $r_{\alpha_{i+1}}^{\prime}$
$(r_{\alpha_{i}}^{\prime}>r_{\alpha_{i+1}}^{\prime})$ such that
$j^{k}G_{\alpha_{i+1},c}(N)\subset O^{\prime}$ for any $c\in
B^{p}(0,r_{\alpha_{i+1}}^{\prime})$. From Lemma 3.1 2, there exists
$\alpha_{i+2}\in\mathbb{N}\setminus\set{\alpha_{1},\ldots,\alpha_{i+1}}$
satisfying $\left\|(\psi\circ g)(q_{\alpha_{i+2}}^{\prime})-(\psi\circ
g)(q_{\alpha_{i+1}}^{\prime})\right\|<\frac{r_{\alpha_{i+1}}^{\prime}}{d-1}$.
Therefore, the case $m=i+1$ holds. ∎

For simplicity, set $I=\set{1,\ldots,d-1}$. By Lemma 3.3 in the case $m=d-1$,
there exist $d$ distinct positive integers $\alpha_{1},\ldots,\alpha_{d}$ and
$d-1$ positive real numbers
$r_{\alpha_{1}}^{\prime},\ldots,r_{\alpha_{d-1}}^{\prime}$
$(r_{\alpha_{1}}^{\prime}>\cdots>r_{\alpha_{d-1}}^{\prime})$ such that for any
$j\in I$,

  1. (d)

$j^{k}G_{\alpha_{j},c}(N)\subset O^{\prime}$ for any $c\in
B^{p}(0,r_{\alpha_{j}}^{\prime})$,

  2. (e)

$\left\|(\psi\circ g)(q_{\alpha_{j+1}}^{\prime})-(\psi\circ
g)(q_{\alpha_{j}}^{\prime})\right\|<\displaystyle\frac{r_{\alpha_{j}}^{\prime}}{d-1}$.

Let $h:N\to P$ be the mapping defined by

$\displaystyle h=\psi^{-1}\circ\left(\psi\circ
g+\sum_{i=1}^{d-1}\rho_{\alpha_{i}}c_{i}\right),$

where $c_{i}=(\psi\circ g)(q_{\alpha_{d}}^{\prime})-(\psi\circ
g)(q_{\alpha_{i}}^{\prime})\in\mathbb{R}^{p}$.

First, we show that $j^{k}h(N)\subset O^{\prime}$. Let $q\in N$ be an
arbitrary point. In the case where $q$ is an element of
$(\bigcup_{j=1}^{d-1}\operatorname{supp}\rho_{\alpha_{j}})^{c}$, since $h=g$
on the open neighborhood
$(\bigcup_{j=1}^{d-1}\operatorname{supp}\rho_{\alpha_{j}})^{c}$ of $q$, we
have $j^{k}h(q)=j^{k}g(q)\in O^{\prime}$. We consider the case where there
exists $j\in I$ satisfying $q\in\operatorname{supp}\rho_{\alpha_{j}}$. Since
$\operatorname{supp}\rho_{\alpha_{j}}\subset\bigcap_{i\in
I\setminus\set{j}}(\operatorname{supp}\rho_{\alpha_{i}})^{c}$ and
$h=G_{\alpha_{j},c_{j}}$ on the open neighborhood $\bigcap_{i\in
I\setminus\set{j}}(\operatorname{supp}\rho_{\alpha_{i}})^{c}$ of $q$, we
obtain $j^{k}h(q)=j^{k}G_{\alpha_{j},c_{j}}(q)$. Moreover, since

(3.12) $\begin{split}\left\|c_{j}\right\|&=\left\|(\psi\circ
g)(q_{\alpha_{d}}^{\prime})-(\psi\circ g)(q_{\alpha_{j}}^{\prime})\right\|\\\
&\leq\sum_{i=j}^{d-1}\left\|(\psi\circ
g)(q_{\alpha_{i+1}}^{\prime})-(\psi\circ
g)(q_{\alpha_{i}}^{\prime})\right\|\\\
&<\sum_{i=j}^{d-1}\frac{r_{\alpha_{i}}^{\prime}}{d-1}\\\ &\leq
r_{\alpha_{j}}^{\prime},\end{split}$

we have $c_{j}\in B^{p}(0,r_{\alpha_{j}}^{\prime})$. The last two inequalities
in 3.12 follow from e and
$r_{\alpha_{j}}^{\prime}>\cdots>r_{\alpha_{d-1}}^{\prime}$, respectively.
Thus, we obtain $j^{k}G_{\alpha_{j},c_{j}}(q)\in O^{\prime}$ by d, which
implies that $j^{k}h(q)\in O^{\prime}$.

Now, we show that $h\in\mathscr{U}_{d}$. For any $i,j\in I$, since
$\rho_{\alpha_{i}}(q_{\alpha_{j}}^{\prime})=\delta_{ij}$ and
$\rho_{\alpha_{i}}(q_{\alpha_{d}}^{\prime})=0$, we obtain

$\displaystyle\left(\psi\circ
g+\sum_{i=1}^{d-1}\rho_{\alpha_{i}}c_{i}\right)(q_{\alpha_{j}}^{\prime})=(\psi\circ
g)(q_{\alpha_{j}}^{\prime})+c_{j}=(\psi\circ
g)(q_{\alpha_{d}}^{\prime})=(\psi\circ h)(q_{\alpha_{d}}^{\prime}),$

where $\delta_{ij}$ is the Kronecker delta. Thus, we have
$h(q_{\alpha_{1}}^{\prime})=\cdots=h(q_{\alpha_{d}}^{\prime})$. Moreover, for
any $j\in I$, the point $q_{\alpha_{j}}^{\prime}$ (resp.
$q_{\alpha_{d}}^{\prime}$) is a topologically critical point of $h$ since
$h=\psi^{-1}\circ(\psi\circ g+c_{j})$ on an open neighborhood of
$q_{\alpha_{j}}^{\prime}$ (resp. $h=g$ on an open neighborhood of
$q_{\alpha_{d}}^{\prime}$). Therefore, we obtain $h\in\mathscr{U}_{d}$.

Finally, we will show that any mapping in $\mathscr{U}$ is not $C^{0}$-stable.
Suppose that there exists a $C^{0}$-stable mapping $\tilde{g}$ in
$\mathscr{U}$. Then, there exists an open neighborhood
$\mathscr{U}_{\tilde{g}}$ of $\tilde{g}$ such that any mapping in
$\mathscr{U}_{\tilde{g}}$ is $C^{0}$-equivalent to $\tilde{g}$. Since we have
shown that $\mathscr{U}_{p+1}$ is dense in $\mathscr{U}$, and topologically
critical points are preserved by homeomorphisms as mentioned in Remark 2.13,
any mapping in $\mathscr{U}_{\tilde{g}}$ has $(p+1)$ topologically critical
points with the same image. This contradicts the fact that the set of all
mappings with normal crossings is dense in $C^{\infty}(N,P)$. $\Box$

## Acknowledgements

The author would like to thank Toru Ohmoto for his kind comments. This work
was supported by JSPS KAKENHI Grant Number JP21K13786.

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$\mathbb{P}*\dot{\mathbb{C}}_{\rho}(\vec{\delta})\ast\dot{\mathbb{S}}_{\gamma}$
forces that $\dot{T}_{\gamma}$ is Aronszajn. However, the last is just
$\star_{\rho}(\vec{\gamma})$, which holds by our inductive assumption and the
fact that $\vec{\gamma}<_{\operatorname{rLex}}\vec{\delta}$. This concludes
the proof of Claim 7.1, which in turn finishes the proof of Theorem 1.1.

We conclude the paper with two questions:

######  Question 7.2.

Is an ineffable cardinal necessary for proving Theorem 1.1? Is a weakly
compact cardinal sufficient?

As we’ve remarked throughout the paper, we only use ineffability in the proof
of Proposition 3.22 (and thus in the corollaries of this proposition). If one
could prove this proposition assuming only a weakly compact, then that would
suffice to show that a weakly compact is optimal.

We also mention the following long-standing question:

######  Question 7.3.

Is a weakly compact cardinal needed for
$\mathsf{SATP}(\omega_{2})+2^{\omega_{1}}=\omega_{3}$?

We recall that in the Laver-Shelah model of $\mathsf{SATP}(\omega_{2})$,
$2^{\omega_{1}}=\omega_{3}$. Moreover, Rinot has shown ([38]) that if the
$\mathsf{GCH}$ and $\mathsf{SATP}(\omega_{2})$ both hold, then $\omega_{2}$ is
weakly compact in $L$.

##  8\. Acknowledgements

The authors would like to thank Itay Neeman, for helpful comments and
corrections, and also the referee, for a thorough reading of the manuscript
and many valuable suggestions and comments.

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