All Questions
5,185 questions
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189
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A weak fixed point property
The usual fixed point property can be interpreted in terms of non empty intersection of the graph of all maps with the graph of the identity map.
This motivates us to consider the following "weak ...
- 366
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1
answer
444
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Stone-Cech compactification of $\mathbb{R}^n$ and smooth functions
I am currently attending a course where we are now covering the Stone-Cech compactification. Today we proved in some detail that extensions of bounded smooth functions on $\mathbb{R}^n$ to $\beta\...
- 19
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1
answer
65
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Hausdorff spaces with asymmetric image relation
For any topological space $(X,\tau)$ we define $$R_{im}(X,\tau) := \{(x,y)\in X^2: (\exists f:X\to X) \text{ continuous and surjective with } f(x) = y\}.$$
Clearly, $R_{im}(X,\tau)$ is reflexive, and ...
- 48.9k
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1
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199
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A countable tight topological group where every countable subset is metrizable
I am looking for an example of a topological group with countable tightness with the property then it is not metrizable, but every countable subset is metrizable but I cannot construct an example.
...
- 987
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3
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172
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Is the poset of all precompact group topologies on an abelian group $G$, order-isomorphic to $\operatorname{Sub}(\hat{G})$?
In this page, in abstract, it is claimed that the poset of all Hausdorff precompact group topologies on an abelian group $G$, is order-isomorphic to the the subgroup lattice of $\hat{G}$, the ...
- 1,145
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1
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386
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Sober topological subspace
Assume $X$ to be a Notherian topological space such that any irreducible closed
subset has a unique generic point. Consider $Y\subseteq X$ as a topological space with the induced topology from $X$. Is ...
- 21
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1
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523
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The space $\psi$
Is the space $\psi$ (described in problem 5I of L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ ...
- 113
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1
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353
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Agreement of two topologies on a linear space
I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.
Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous ...
- 8,512
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1
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310
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A question from Arhangel'skii-Buzyakova
The question is also posted here, however there is no answer.
Recently, I am reading the paper: On linearly Lindelöf and strongly discretely Lindelöf spaces by Arhangel'skii and Buzyakova. Here is ...
- 654
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1
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79
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Does the network of $X$ equal to the network of $C_p(X)$?
Does the network of $X$ equal to the network of $C_p(X)$?
$C_p(X)$ denotes the set of all real-valued continuous functions on $X$ endowed with the topology of pointwise convergence.
Thanks!
- 654
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2
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407
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What are Normal Sets (Fréchet)?
In 1913, LEJ Brouwer started a new approach to give a topologist's definition of the notion dimension ("Über den natürlichen Dimensionsbegriff", Journal für die reine und angewandte Mathematik, 142, ...
- 11
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1
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566
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Example of a topological space
In my recent research, I defined a topological space $X$ to be an $EZ$-space if for every open subset $A$ of $X$, there exists a collection $\{A_{\alpha}: \alpha\in S\}$ of clopen subsets of $X$ such ...
- 192
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1
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136
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Nonhomeomorphic CW-complexes that are "stably" homeomorphic
Do there exist CW-complexes $X$ and $Y$ that are not homeomorphic, but $X \times I$ and $Y \times I$ are homeomorphic? Here $I$ denotes the unit interval $[0, 1]$.
- 19
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1
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752
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3D surfaces of infinite genus
How might one show that the set of connected 3D surfaces with infinite genus (up to homeomorphism) is countably infinite?
We could either use proof by contradiction or come up with a way to count ...
- 11
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1
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1k
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Maximal ideals and ultrafilters [closed]
I am not sure about these two definitions.
For example, if we take the power set of A={1,2,3} with the partial order of inclusion. What are the maximal ideals and what are the maximal filters? For ...
- 11
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1
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107
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Approximate selection theorems for factoring through perfect maps
I have the following setup:
$X, Y$ are topological spaces (if it helps, they can both be $T_1$ and normal. They can even be countably paracompact. They can't be assumed paracompact). $V$ is a normed ...
- 1,331
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1
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162
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Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
It seems too good to be possible, but:
Is there a uniformly continuous injective image of $(0,1)\setminus\Bbb Q$ in the Cantor space?
Here, the Cantor space $\{0,1\}^{\Bbb N}$ is equipped with the ...
- 3,104
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1
answer
153
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For topological torus action, there is a subcircle whose fixed point is the same as the torus
Let $T=\mathbb{S}^{1}\times \mathbb{S}^{1}\times \cdots \times \mathbb{S}^{1}
$ ($n$ times) be an $n$-dimensional torus acting on any topological space $X$.
The group $G$ is said to act on a space $X$ ...
- 1,367
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1
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142
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Congruences that aren't "finite from above," take 2: semigroups
This is a hopefully less trivial version of this question. Briefly, say that a congruence is parafinite if it is the largest congruence contained in some equivalence relation with finitely many ...
- 20.5k
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628
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Cohomology of the amplitude space of unlabeled quantum networks
I am investigating a particular map from a product of three-spheres to the moduli space of (non-negative, real edge weight) networks. The map in question is
$$f: \smash{\left( \mathbb{S}^3 \right)}^N \...
- 571
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2
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248
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A few questions about Tychonoff plank
In the Morita's following article (K. Morita. Some properties of M-spaces), constructing an space $X$ and defining an identification on it.
My first question is how to prove that $S$ is countably ...
- 1,367
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1
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81
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Is the class of rc-spaces closed under products?
Let $(X,\tau)$ be a topological space. A retraction is a continuous map $r:X\to X$ such that $r$ is the identity on $\text{im}(r)$. We call $S\subseteq X$ a retract of $X$ if there is a retraction $r:...
- 48.9k
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204
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Name of a space with both a topology and a metric that are not compatible?
Let $(X,\tau,d)$ be a space where $\tau$ is a topology and $d$ is a metric, where the topology $\tau$ is not necessarily compatible with $d$.
Is there a canonical name for such a structure (maybe ...
- 775
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135
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Annulus theorem for pseudomanifolds
Lets say I take an arbitrary closed and smooth $d$-manifolds $\mathcal{M}$. Now, it is a well-known fact that whenever I take two (sufficiently nice embedded) closed $d$-balls $B_{1}$ and $B_{2}$ in $\...
- 1,429
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90
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Affine semigroup generating a lattice
This is a cross-post from MSE.
Everything is assumed to be finite-dimensional. Let $S$ be a finitely generated affine semigroup (i.e. a subsemigroup of a lattice $N$ of a Euclidean space). Assume that ...
- 1,374
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1
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152
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Is the Isbell–Mrówka space $\Psi(\mathcal A)$ with $\lvert\mathcal A\rvert=\omega_1$ starcompact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $\operatorname{St}(\bigcup\mathcal V,\mathcal U)...
- 505
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117
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Does there exist a starcompact space which is not star-$K$-compact?
A space $X$ is said to be starcompact if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V\subseteq\mathcal U$ such that $St(\cup\mathcal V,\mathcal U)=X$.
A space $X$ ...
- 505
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1
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323
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Is the restriction of a projection to a compact subset a quotient map?
Let $(X, \mathcal{T}_X)$ and $(Y, \mathcal{T}_Y)$ be topological spaces, $Z = X \times Y$, $\mathcal{T}_Z$ be the product topology on $Z$, $f : Z \to X$ be defined by $f(x, y) = x$, and $C \subset Z$ ...
- 397
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1
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233
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An approximation property in a separable topological vector space
Let $X$ be a topological vector space.
Let us say that $X$ enjoys sequential separablity if there exists a sequence $\{x_n\}$ in $X$ such that for every $x\in X$ there exists a subsequence of $\{...
- 4,058
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1
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82
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Does there exist a strongly star-Lindelöf space which is not DCCC?
A space $X$ is said to be strongly star-Lindelöf if for every open cover $\mathcal U$ of $X$ there exists a countable subset $A$ of $X$ such that $St(A,\mathcal U)=X$.
A space $X$ has discrete ...
- 505
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1
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156
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Necessary and sufficient conditions for the Lie group embedding $G \supset J$ can be lifted to $G$'s covering space [closed]
Suppose the Lie group $G$ contains the Lie group $J$ as a subgroup, so
$$
G \supset J.
$$
If $G$ has a nontrivial first homotopy group $\pi_1(G) \neq 0$.
If $G$ has a universal cover $\widetilde{G}$, ...
- 317
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1
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88
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Subsets of $\mathbb{R}$, every nonempty subset of which generates a disconnected translation-invariant topology
Let $\mathbb{R}$ be the set of real numbers. Given a subset $S$ of $\mathbb{R}$, let $\mathcal{T}_S$ be the translation-invariant topology generated by $S$. That is, $\mathcal{T}_S$ is the topology ...
- 175
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1
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650
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Local diffeomorphisms, covering maps and smooth path lifting
Let $f: M\to N$ be a surjective local diffeomorphism of noncompact smooth manifolds.
Suppose that every smooth path is liftable, that is, for any smooth path $\gamma: [0,1]\to N$ and any point $p\in f^...
- 2,934
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1
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141
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Action of the permutation group on the set of topologies on a continuum
Let $X$ be a set of continuum cardinality. The group of permutations of $X$ acts on the set of topologies on $X$.
What can be said about the fixed points of this action? Are manifolds fixed points?
...
- 13
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1
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261
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CH and the density topology on $\mathbb{R}$
In the article AN EXAMPLE INVOLVING BAIRE SPACES (https://www.ams.org/journals/proc/1975-048-01/S0002-9939-1975-0362249-1/S0002-9939-1975-0362249-1.pdf) of H. E. White Jr. it is shown that, assuming ...
- 561
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1
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227
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Is a topology sandwiched between two norms compactly generated?
Recall that a Hausdorf topological space $X$ is called compactly generated if any set whose intersections with compacts are compact is closed. Locally compact and first countable spaces are compactly ...
- 5,529
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207
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Boundary value of a homeomorphism
Assume that $f$ is a homeomorphism of the unit disk onto itself. Assume also that $f$ has a continuous extension up to the boundary. It seems that $f$ is monotonous at the boundary, but I need a ...
- 49
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208
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Is every homeomorphism approximately a product of homeomorphisms?
Let $\phi$ be a homeomorphism on $\mathbb{R}^{n+m}$, $\epsilon>0$, and $K\subseteq \mathbb{R}^n$ be a non-empty compact. Does there necessarily exist homeomorphisms $\phi_1,\phi_2$ on $\mathbb{R}^...
- 5,405
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346
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Comparison of topology of pointwise convergence and compact-open topologies for Sierpiński space
Let $\{0,1\}$ be equipped with the Sierpiński topology $\{\emptyset, \{0,1\},\{1\}\}$, and $\mathbb{R}^d$ with the usual Euclidean topology. Then is the pointwise-convergence (point-open) topology ...
- 5,405
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295
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For pseudo-Cauchy sequence, does there always exist a pseudo-Cauchy subsequence of $(x_n)$ having distinct terms?
Given a metric space $(X,d),$ let us call a sequence $(x_n)$ in $X$ to be pseudo-Cauchy if $\lim\limits_{n\to\infty}d(x_n,x_{n+1})=0.$
For example, the sequence $\left(1+\frac{1}{2}+\frac{1}{3}+\...
- 195
1
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1
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234
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Quotients of the irrationals
Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
- 3,317
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1
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160
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Comparing $(((A^\varepsilon)')^\varepsilon)'$ and $int(A)$, where $A' := X\setminus A$ and $A^\varepsilon := \{x \in X \mid d(x,A) \le \varepsilon\}$
Disclaimer. This is follow up to the question https://math.stackexchange.com/q/3486130/168758.
Let $X=(X,d)$ be a Polish metric space equiped with the Borel $\sigma$-algebra and let $\mu$ be a ...
- 6,853
1
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1
answer
96
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Degree of continous function, a question about its representation
Let $f \in C(\mathbb R,\mathbb R)$, $\text{degree}(f)=\sup\limits_{a \in\mathbb R} \{ \text{card}(f^{-1}(\{a\})) \}$
Is it true that $\forall f \in C(\mathbb R,\mathbb R),\text{degree}(f)=k\in\mathbb ...
- 4,074
1
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1
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88
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closed connected subspace of a cartesian product
Let $Y$ be a connected CW-complex and $F\subset Y\times Y$ be a closed connected subspace such that the composition $F\subset Y\times Y \rightarrow Y$ is a bijective map, where
$Y\times Y\rightarrow ...
- 223
1
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1
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192
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$\mathbb{R}$-like spaces [closed]
Let us call a topological space $(X,\tau)$ $\mathbb{R}$-like if it is homogeneous, connected, $T_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$.
What is an example of an $...
- 48.9k
1
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1
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192
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Countable union of Menger spaces
A topological space $X$ has Menger's property $\textsf{S}_{\mbox{fin}}(\mathcal{O}, \mathcal{O})$ if, for each sequence of open covers, $\mathcal{U}_1, \mathcal{U}_2, \cdots $, we can select finite ...
- 561
1
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1
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1k
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Countable intersections in topological space
If a T1 topological space is closed under countable intersections, does this necessarily make the topology discrete? It is easy to construct a counterexample if the topological space is not assumed to ...
1
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3
answers
408
views
Extension of continuous map on metric space
Let $X$ be a compact metric space, $A\subset X$ a closed subset and $f:A\to A$ be a continuous map.
Can $f$ be extended to a continuous map $X\to X$?
If so, is there an extension which is injective if ...
user130903
1
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1
answer
118
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Is there a homeomorphism between the sets of Schur stable and Hurwitz stable matrices in companion forms?
This is a cross-post to the question I asked at MSE.
The set of Schur stable matrices is
\begin{align*}
\mathcal S = \{A \in M_n(\mathbb R): \rho(A) < 1\},
\end{align*}
where $\rho(\cdot)$ denotes ...
- 195
1
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1
answer
58
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A sequence in generalized order spaces
Let $X$ be a GO-space with the topology $\tau$ and $\lambda$ be the usual open interval topology on $X$. Put
$$ R= \{x\in X: [x, \rightarrow) \in \tau\setminus \lambda \} \text{ and } L= \{x\in X: (\...
- 89