All Questions
5,185 questions
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A Borel perfectly everywhere surjective function on the Cantor set
Does there exist a Borel (or even continuous) function $f:\mathcal{C}\to\mathcal{C}$, where $\mathcal{C}$ is the Cantor set (or Cantor space $2^\omega$) such that for every nonempty closed perfect set ...
- 3,115
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1
answer
108
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The cardinal of the closure of a set in a topological space
Suppose $E$ is a Hausdorff topological space, $A$ is a subset of $E$. Card(A) is the cardinal of $A$. $\bar{A}$ is the closure of $A$. Do we have
$Card(\bar{A})\le 2^{2^{Card(A)}}$.
- 11
1
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1
answer
231
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Continuous semigroup homomorphism of composition to additive structure
Let $G$ be the topological semigroup whose underlying space is $C(\mathbb{R}^d,\mathbb{R}^d)$ equipped with composition as semigroup operation and let $H$ be the topological group whose underlying ...
- 5,405
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2
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1k
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Understanding reduced suspension of $S^1$ [closed]
I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product ...
- 71
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1
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158
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Trouble with plane embedding
Let $C$ be the middle-thirds Cantor set. Obviously $C\times [0,1]$ embeds into the plane. But $C\times D$ does not, $D$ being a closed disc in the plane.
Are there any general results which can be ...
- 3,317
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1
answer
141
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Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable?
A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.
Let $X$ be a Lindelof, perfectly normal, $\sigma$-space.
Must $X$ be separable?
Thanks very much.
- 621
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3
answers
304
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Topological properties via properties continuous maps [closed]
A topological space $(X,\tau)$ is connected if and only if the only continuous maps $f:X\to\{0,1\}$ (where $\{0,1\}$ carries the discrete topology) are the constant maps.
Are there other examples of ...
- 48.9k
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2
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207
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Smooth, irreducible surface with real part containing two projective planes
Let $X$ be a smooth and irreducible projective variety over $\mathbb{R}$ of dimension two. I am looking for an instance of such a variety where two distinct connected components of $X(\mathbb{R})$ are ...
- 3,031
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1
answer
284
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Hausdorff spaces such that every subset is a retract
Let $(X,\tau)$ be a Hausdorff space such that for every non-empty $A\subseteq X$ there is a continuous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$. Does $\tau$ have to be discrete?
- 48.9k
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1
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171
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US does not imply AB
We say that a topological space $X$ is:
$AB$, provided that $X$ is $T_1$ and for each pair $(A, B)$ of compact, disjoint subsets of $X$ there is $U$, an open subset of $X$, such that either $A \...
- 11
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2
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411
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Existence of non-locally constant functions
Given a nondiscrete compact Hausdorff space $K$, does there always exist a real-valued function $f$ on $K$ that is not locally constant? Why/why not?
In http://arxiv.org/abs/math/9505204 the authors ...
- 388
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1
answer
244
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Finding a good ordering of $\mathbb{Q}$
Oftentimes in density arguments we let $\{x_n\}$ be a dense sequence and this is sufficient to imply the desired result.
From a research question I am working on I have simplified the example/...
- 882
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2
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166
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TSP, but for all routes not all points
Hello
I am trying to solve a problem involving a cross-country ski trail map. I wish to travel every trail on the map, at least once, but no more than twice (so I can out-and-back on a destination, ...
- 113
1
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1
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997
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An open problem on general topology
There is an open problem in this paper: Classes defined by stars and neighbourhood assignments by van Mill and others.
Problem 4.8. Is a regular (Tychonoff) star compact space metrizable if it has a $...
- 654
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1
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582
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Lifting identities of formal power series
I am looking for a possibly general class of algebraic structures (maybe special topological rings) in which one can deduce identities of concrete power series from formal ones. This class should ...
- 13
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1
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636
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Does anyone know an example of non-separable $L^1$ of a probability space?
It is easy to give examples of non-separable $L^p$ spaces by considering a measure on a big space. If one adds the condition that the space has to have total measure 1, the problem is not as easy.
...
- 96
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1
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2k
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On Zariski Dense Subsets
Can we find a Zariski-dense subset $U$ of the affine plane over the complex number field, such that any subset $U^{\prime}$ of $U$ has its closure either the whole affine plane or finite number of ...
- 211
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1
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608
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About deformation retract
Let $X\subset Y$ be CW-complexes. Denote $i\colon X\to Y$ be an inclusion map.
Is it true that $i$ is deformation retract if and only if $i$ is homotopy equivalence?
When I saw some papers about h-...
- 13
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2
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378
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Is this a pre-ordered commutative semigroup?
Motivation
I'm studying an approach to axiomatic thermodynamics based on the notion of commutative semigroup $(S,+)$ with a preorder relation $\to$ on $S$. In other words, $S$ is non-empty set, the ...
- 33.3k
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1
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130
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A finite dimensional continuum with a subset $A$ such that both $A$ and $X\setminus A$ are dense and contractible
Inspired by this question we ask the following question.
Note that the comment conversation and answers to the above question imply that
There are two complementary subsets of the unit ...
- 366
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2
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132
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Description of atomless complete Boolean algebras with a countable $\pi$-base
Recall that a subset $A$ of a Boolean algebra $B$ is a $\pi$-base if for every $b>0$ there is $a\in A$ with $0<a\le b$. For example, the definition of atomicity says that atoms constitute a $\pi$...
- 5,529
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344
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Is there anyway to formulate the Alexandrov topology algebraically?
One knows that the Alexandrov topology on a preordered set is the finest topology that induces the same [specialization] preorder on the set.
Given this, one finds a one-to-one correspondence between ...
- 612
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1
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119
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Extremally disconnected rigid infinite Hausdorff compacta(?)
Question: does there exist an extremally disconnected infinite Hausdorff compact space $\ X\ $ such that the only homeomorphism
$\ h: X\to X\ $ is the identity homeomorphism
$\ h=\mathbb I_X:\ X\to X\...
- 4,786
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1
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732
views
Notations for open and closed sets
I am wondering why a standard notation for open sets is $G$ and that for closed sets is $F$. I mean, $F$ precedes $G$ in the alphabet, whereas open sets are usually introduced before closed ones.
- 128k
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360
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A question about realcompact spaces
Let $X$ be completely regular space, $\beta X$ be Stone-Čech
compactification of $X$, and $\upsilon X$ be Hewitt realcompactification of $X$.
Then $X\subset \upsilon X\subset \beta X$.
If the ...
- 1,367
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1
answer
250
views
When are fixed point sets in $T_1$ spaces always closed?
Let $X$ be a topological space, and say that $X$ satisfies the closed fixed point set property if every continuous self-map $f:X\to X$ has fixed point set $\operatorname{Fix}(f)=\{x\in X\mid f(x)=x\}$ ...
- 2,821
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258
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Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...
- 113
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149
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Does there exist a star-Lindelöf space which is not DCCC?
A space $X$ is said to be star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $\mathcal V$ of $\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal ...
- 505
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2
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545
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Subsets of the Cantor set
A copy of the Cantor set is a space homeomorphic to $2^{\omega}$.
Suppose that $X$ is a Hausdorff space that contains a copy $C^{\prime}$ of the Cantor set. Let $U$ be a nonempty subset open in $C^{\...
- 31
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248
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The Schoenflies Theorem on two dimensional surfaces
Let $S$ be a surface and $U$ an open connected subset of $S$. If the frontier of $U$ in $S$ is a two sided circle $C$, then the closure of $U$ in $S$ is a surface whose boundary is $C$. I would like ...
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1
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143
views
Is the Rudin-Keisler ordering a continuous relation?
If $X, Y$ are topological, and $R\subseteq X\times Y$ we say that $R$ is continuous (from $X$ to $Y$) if for every $V\subseteq Y$ with $V$ open, we have $$R^{-1}(V) = \{u\in U: \exists v\in V:(u,v)\...
- 48.9k
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925
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Known dense subset of Schwartz-like space and $C_c^{\infty}$?
After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
- 5,405
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1
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284
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Poincare duality-differential geometry
Let $ M $ be a smooth and compact manifold with boundary $\partial M = X \times F $ on which the structure of a smooth locally trivial bundle $$ \pi: \partial M \longrightarrow X $$
where the $ X $ ...
- 27
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2
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176
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Connected Hausdorff spaces with large collection of disjoint open sets
Is there for every infinite cardinal $\kappa$ a connected Hausdorff space $(X,\tau)$ with $|X| = \kappa$ and a collection ${\cal D}$ of mutually disjoint open sets with $|{\cal D}| = \kappa$?
- 48.9k
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1
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136
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A permutation group inducing a topologically transitive action without dense orbits on $\omega^*$
Let $G$ be a subgroup of the permutation group $S_\omega$ of the countable infinite set $\omega$. Each bijection $g\in G$ admits a unique extension to a homeomorphism $\bar g$ of the Stone-Cech ...
- 42k
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1
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167
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Collineations of projective spaces and isomorphisms of fields
For a (topological) field $F$ by $FP^2$ we denote the projective plane, i.e., the quotient space of $F^3\setminus\{0\}^3$ by the equivalence relation $\vec x\sim\vec y$ iff $\vec x=\lambda\vec y$ for ...
- 42k
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2
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195
views
Reference request: lower sets of a preorder form a lattice
Consider a set $S$ with a preorder $\preceq$ (a preorder is a reflexive and transitive relation). A lower set $A$ of $S$ is defined as a subset of $S$ such that for all $x \in S$ and $y \in A$, if $...
- 695
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2
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154
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Requirement for connected sets [closed]
Let $E$ be a compact metric space. Suppose that closure of every open ball $B(a,r)$
is the closed ball $B'(a,r)$. Must every open ball in $E$ be connected?
I think it most probably is. But I don't ...
- 11
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1
answer
104
views
Is the map between mapping spaces, induced by the functor $\vert Sing(-)\vert$ continuous?
Let $X$ and $Y$ be topological spaces. Let $\vert Sing(-)\vert$ be the functor which sends a topological space to the (or "a"? there seem to be more possibilites, for me it's just important, that I ...
- 181
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92
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Does this collection of properties imply metrizable?
I am working with a space with the following properties, and want to know if it is necessarily metrizable.
countable union of compact nowhere dense sets
T$_4$$=$T$_1$+normal
separable
Lindelöf
I ran ...
- 3,317
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1
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106
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Which positive integers can occur as the genus of a numerical semigroup minimally generated by 3 (or 2) elements?
Let $S$ be a numerical semigroup. Let $g(S)=|\mathbb N \setminus S |$, where $\mathbb N$ here denotes the set of non-negative integers. Let $e(S)$ be the embedding dimension of $S$, i.e. the ...
user111492
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2
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223
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Is $C_b(Q,E)$ linearly isometrically isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone–Čech compactification of $Q$?
Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space.
Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ and $C_b(Q,E)$ be the collection of all $E$-valued ...
- 623
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284
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Cantor set onto connected set?
Let $X$ be a Hausdorff space such that the irrationals $\mathbb P$ (in their usual topology) form a dense subspace of $X$.
Let $C$ be the Cantor set. The set of "non-endpoints" of $C$ is ...
- 955
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2
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473
views
How to choose a continuous function which vanishes **only** on the closed set
We are reading John Roe's book Lectures on Coarse Geometry. We come across a question in P27 line 9:
Suppose $X$ is a paracompact and locally compact Hausdorff space, $\bar{X}$ is a ...
- 135
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1
answer
79
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Products of spaces with an underlying free ultrafilter as topology
If ${\cal U}$ is any ultrafilter on $\omega$, the pair $(\omega,{\cal U}\cup \{\emptyset\})$ is a connected topological space. Is there a non-principal ultrafilter ${\cal U}$ on $\omega$ such that we ...
- 48.9k
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1
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130
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Two consecutive continua
Are there two non homeomorphic continua $X,Y$ such that $X $ can be embedded in $Y$ but there is no topological space $Z$ with $$X<Z<Y.$$
The later relation means that $Z$ ...
- 366
1
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1
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89
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Name for a certain topology on boundary points of convex sets
I was looking through some old notes of mine and stumbled upon a question I had wanted to ask a while back but never got around to it. Here it is now:
Consider a convex set $X\subseteq \mathbb{R}^n$ ...
- 9,650
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1
answer
134
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a space isomorphic to $S^{p+q}$
I've asked this question few days ago in MathStackExchange, but there was no result. So, I decided to ask it there.
In one of the paper I have met that
$$\mathbb{S}^{p+q} \cong \mathbb{S}^p \times \...
- 195
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1
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160
views
Is the following set closed with respect to the Hausdorff metric? [closed]
Let $(X,d)$ be a non-empty complete metric space, let
M be the set of all non-empty compact subsets equipped
with the Hausdorff metric, and $N$ be a positive integer.
Is
$$
\{A\subset X : 1\le \# A \...
- 244
1
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1
answer
212
views
Identifying attractors in high dimensional dynamical sytems [closed]
I have a high dimensional dynamical system, and I was wondering if there is a method to identify the various attractors of the system i.e, a way of mapping the energy landscape?
I was thinking of a ...
- 19