**0**

votes

**0**answers

16 views

### Dirac functional embedding

I got the following set up:
Let $S \neq \emptyset$ equiped with the discrete topology and let $\ell_\infty(S) = \{f: S \to \mathbb C \mid f \text{ bounded}\}$. Not $\ell_\infty(S)$ with the pointwise ...

**1**

vote

**0**answers

14 views

### Bernstein sets of large cardinality

A metrizable space $X$ will be called a generalized Bernstein set is every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$.
It is well-known that the real line contains ...

**7**

votes

**0**answers

79 views

### What is the smallest density of a metrizable space without countable separation?

A Tychonoff space $X$ is defined to have countable separation if some (equivalently, any) compactification $bX$ of $X$ contains a countable family $\mathcal U$ of open sets such that for any points $x\...

**0**

votes

**1**answer

79 views

### Is the following set closed with respect to the Hausdorff metric? [on hold]

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 \...

**1**

vote

**2**answers

91 views

### Topological properties via properties continuous maps

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 ...

**5**

votes

**1**answer

273 views

### Does the topology induced by the Hausdorff-metric and the quotient topology coincide?

Assume that $X$ is a metric space, and $\sim$ is an equivalence relation on $X$.
Furthermore we assume that the number of elements in each equivalence class
is bounded by a positive constant.
Does ...

**2**

votes

**1**answer

93 views

### Spaces $Y$ such that $C(-, Y)$ is always acceptable

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to ...

**-1**

votes

**0**answers

81 views

### Open set in $\mathbb{C}\times \mathbb{C}$ [closed]

Let $T_{z}, z\in\mathbb{C}$ an unbounded operator with domain $D$ subspace of a Hilbert space $H$ onto $H$, We assume $T_{z}$ holomorphic in $z$.
Let $R(\xi,z)=(T_{z}-\xi)^{-1}$ its resolvent where $\...

**3**

votes

**1**answer

147 views

### Reference or counter-example for Closed Graph Theorem for multivalued maps in general topological spaces

Could someone be so kind to point me in the direction of a citeable proof of the following version of the Closed Graph Theorem? (i.e. assuming this is true, could someone give me a literature ...

**-4**

votes

**0**answers

73 views

### generalized $F_{\sigma}$-sets [closed]

Let $(X,\tau)$ a topological space and $S\subset X$.
$S$ is a generelized $F_{\sigma}$-set if $\forall G\in \tau$ such that $S\subset G$ exists an $F_{\sigma}$-set $F$ such that $S\subset F\subset G$....

**3**

votes

**0**answers

141 views

### Topology on $\mathcal{C}(X,Y)$ to work with homotopy

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...

**2**

votes

**1**answer

91 views

### Admissible and proper topologies on $C(X,Y)$

Given non-empty sets $A, B, C$, set $B^A$ to be the set of all functions $f:A\to B$ there is a natural bijection $\Lambda: C^{A\times B} \to (C^A)^B$ defined in the following way: for $f:A\times B \to ...

**12**

votes

**2**answers

419 views

### Which spaces have enough curves

Let $\mathbf{Top}$ be the category of topological spaces, and let $I\in\mathbf{Top}$ be the unit interval $I=[0,1]\subset\mathbb{R}$. For any space $X$, let $|X|$ denote the underlying set of points; ...

**5**

votes

**0**answers

63 views

### Set of w*-continuous operators closed for the weak* topology or not?

Let $X$ be a dual Banach space, i.e. $X=(X_*)^*$ for some Banach space $X_*$. Consider the weak* topology of $B(X)$, i.e. the topology of pointwise convergence on $X$ endowed with the $\sigma(X,X_*)$-...

**3**

votes

**2**answers

432 views

### What is a generalized limit?

In the proof of Lemma 1.3 in the paper "The ideal structure of a groupoid C* algebra", Journal of Operator Theory 1991 by Jean Renault, I found the notion of a generalized limit of a net without any ...

**-1**

votes

**0**answers

12 views

### Continuous function on compact topological space [migrated]

I came across the following statement.
Let $X$ a uncountable set, $p \notin X$ and $X^* = X \cup \{p\}$. Let $$\mathcal O := \{O \subseteq X^* \mid O \subseteq X \text{ or } p \in O \text{ and } X \...

**2**

votes

**2**answers

73 views

### A contractible non-planar continuum

Let $Z=\{0\}\cup\{\pm\frac1n\}_{n\in\mathbb N}$ be the sequence that converges to zero from both sides. Consider the contractible continuum $$A=(Z\times[-1,1]\times\{0\})\cup([-1,1]\times\{0\}\times\{...

**0**

votes

**0**answers

107 views

### A topology on the product space of Euclidean space and smooth functions space

I'd like to know if there is a well-known topology on the space $S := \mathbb R \times C^\infty(\mathbb R)$, such that $(x_n, f_n) \to (x, f)$ in $S$ with respect the topology is equivalent to
$$(x_n,...

**2**

votes

**1**answer

85 views

### Neighborhoods with proper multiplication

The following question was originally asked here, by C. Dubussy: http://math.stackexchange.com/questions/1802111/neighbourhoods-with-proper-multiplication
Assume we have two closed subsets $F$ and $G$...

**5**

votes

**0**answers

172 views

### Topology on the space of Borel measures

Let $ B $ be the set of all measures $ \phi $ of $ \mathbf{R}^{n} $ such that every open set is $ \phi $-measurable (sometimes these measures are called Borel measures). Note the measures in $ B $ are ...

**-4**

votes

**0**answers

19 views

### If a function is continuous in one chart, it is continuous in every chart of a maximal atlas of a topological manifold? [migrated]

This is a correct statement, can somebody show how to prove it specifically?
I am not quite sure about what it means by 'If a function is continuous in one chart ...', in my understanding, a chart is ...

**2**

votes

**1**answer

84 views

### An example of a regular but not well-based topological space

Call a topological space $\langle X,\mathscr{O}\rangle$ regular iff it is both $T_0$ and $T_3$: for every point $x\notin A$, where $A$ is a closed subsets of $X$, there are open and disjoint sets $V$ ...

**1**

vote

**1**answer

70 views

### Metrizability of the space of probability measures endowed with the topology of setwise convergence

Let $X$ be a separable completely metrizable space, let $\mathscr{B}(X)$ denote the Borel $\sigma$-algebra on $X$, and let $\mathscr{P}(X)$ denote the space of all probability measures on $(X, \...

**-4**

votes

**2**answers

178 views

### Continuous map from $\mathbb R^2$ to $\mathbb R$? [closed]

There must be a map from $\mathbb R^2$ to $\mathbb R$, since they are the same cardinality. But is there a construction for a continuous map from $\mathbb R^2$ to $\mathbb R$?
I guess what I mean is ...

**1**

vote

**0**answers

65 views

### Do $G_\delta$-measurable maps preserve dimension?

This question (in a bit different form) I leaned from Olena Karlova.
Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...

**0**

votes

**0**answers

62 views

### Reference request: Uniformly totally bounded classes of compact metric spaces are Gromov-Hausdorff precompact

The following Theorem can be found for instance here (Theorem 7.4.15):
Theorem. (author ?) Any uniformly totally bounded class $\mathfrak X$ of compact metric spaces is pre-compact in the Gromov-...

**7**

votes

**1**answer

173 views

### Product of limit $\sigma$-algebras

Let $X$ and $Y$ be Polish (i.e. Borel subsets of separable completely metrizable) spaces. For a Polish space $Z$, let $\mathscr{S}(Z)$ denote the limit $\sigma$-algebra on $Z$, i.e. the smallest $\...

**7**

votes

**1**answer

259 views

### Abstract result on partitions of unity?

A motivation: The classical Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the abstract Stone-Weierstrass theorem (and also the ...

**2**

votes

**2**answers

286 views

### Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...

**2**

votes

**0**answers

45 views

### Zero-dimensional $F$-space which is not strongly zero-dimensional

Does anyone know of an example of a (Tychonoff) $F$-space which is zero-dimensional but not strongly zero-dimensional?
By an $F$-space we mean every cozeroset is $C^*$-embedded.
By zero-dimensional ...

**1**

vote

**0**answers

47 views

### Dual equivalence for multioperators

This is a reference request question. But let's start with a few definitions.
Let $L$ and $M$ be two bounded lattices. A multioperator $p$ for $L$ and $M$ is an application $$p : L \to Ft(M)^{op}$$ ...

**4**

votes

**2**answers

494 views

### Some examples of clean topological spaces

I asked this question at MSE but I did not received any answer, so I repeat it here at MO:
What is an example of a Hausdorff topological space $X$, not a singleton, such that the ring $C(X)$...

**6**

votes

**0**answers

117 views

### Is every Polish ring topology on $\mathbb{C}$ defined by an absolute value?

There is a unique up to isomorphism algebraically closed field of characteristic 0 and cardinality of the continuum. Let's call it $K$.
We usually call it $\mathbb{C}$, but by this we impose a ...

**4**

votes

**0**answers

183 views

### Unbounded towers and combinatorial cardinal characteristics of the continuum

Update: Perhaps the question is too difficult. I would appreciate, thus, even just comments or related observations.
This question assumes familiarity with combinatorial cardinal characteristics of ...

**28**

votes

**1**answer

724 views

### What is the meaning of this analogy between lattices and topological spaces?

Let me add one more edit to help explain why this is a serious question. Theorem 5 below is a sort of lattice version of Urysohn's lemma, and it has essentially the same proof. Theorem 6, the famous ...

**0**

votes

**0**answers

74 views

### Right split for homomorphism onto $S_\infty$

Let $G$ be a closed subgroup of $S_\infty$ and let $f:G\rightarrow S_\infty$ be a continuous surjective homomorphism. Under which conditions $f$ has a right split, i.e. there exists some $g:S_\infty\...

**1**

vote

**0**answers

131 views

### $\mathbb E$-descent maps in topological spaces in terms of different sites?

The paper Facets of Descent I by Janelidze and Tholen defines $\mathbb E$-descent maps as those for which $\Phi^p:\mathbb EB\longrightarrow \mathsf{Des}_\mathbb{E}(p)$ is an equivalence of categories.
...

**5**

votes

**1**answer

300 views

### When is there an unbounded tower in $[\mathbb{N}]^\infty$?

(Edit: I'm splitting the question, leaving here only what is answered by Ashutosh, and moving the rest to another question.)
This question assumes familiarity with combinatorial cardinal ...

**12**

votes

**1**answer

527 views

### Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way:
Let $X$ be a set and let $\mathcal{P}...

**7**

votes

**1**answer

208 views

### What is the height (or depth) of $[\mathbb{N}]^\infty$?

(This question assumes familiarity with combinatorial cardinal characteristics of the continnum.)
Let $[\mathbb{N}]^\infty$ be the family of infinite subsets of $\mathbb{N}$,
partially ordered by $\...

**1**

vote

**0**answers

60 views

### The role of absolute continuity in stochastic ordering defined over sets of probability distributions

This question is about a claim given in this paper (page 261, the remark), but without any proof.
It simply says that if two sets of probability distributions, $\mathscr{P}_0$ and $\mathscr{P}_1$ (...

**4**

votes

**1**answer

110 views

### Minimal zero-dimensional Hausdorff spaces

A topological space $(X,\tau)$ is said to be zero-dimensional Hausdorff (zdH) if for $x\neq y\in X$ there is $C\subseteq X$ clopen (closed and open) such that $x\in C$, but $y\notin C$.
We say a zdH ...

**7**

votes

**2**answers

339 views

### Link between the hairy ball theorem and the fundamental theorem of algebra

I read in the book "Concepts of modern mathematics" by Ian Stewart that it was possible to proof the fundamental theorem of algebra using the hairy ball theorem (complete reference to the page is in ...

**0**

votes

**0**answers

53 views

### Fixed point shape property

Question: Provide (or prove that it's not possible) a metric compact space which has the fixed point property but not the fixed point shape property.
Here is the definition of f.p.s.p.("map" means ...

**2**

votes

**1**answer

105 views

### Surniversal spaces

Basic background
On one hand there is a complete result: $\,\ $for every non-negative integer $n$ there exists an $n$-dimensional compact metric space $M^n$ such that it contains a homeomorphic ...

**2**

votes

**1**answer

69 views

### Topology with no direct lower neighbor

Given any poset $(P,\leq)$ and $x, y\in P$ we set $[x,y] = \{p\in P: x\leq p \leq y\}$. For any set $X$, let $\text{Top}(X)$ denote the set of topologies on $X$. The set $\text{Top}(X)$ is a complete ...

**1**

vote

**1**answer

194 views

### Applications of topology to discrete dynamical systems?

I'd like to know some of the applications of topology to discrete dynamics. By discrete dynamics I loosely mean studying maps between discrete sets.
I mean cases where adding a topology to the sets ...

**3**

votes

**1**answer

187 views

### Fréchet L-Spaces

According to the paper The emergence of open sets, closed sets, and limit points
in analysis and topology famous mathematician Maurice Fréchet who introduced the concept of metric spaces has also ...

**2**

votes

**1**answer

242 views

### Local “pathologies” in spaces arising naturally in algebraic topology

I have been thinking about methods for constructing continuous paths locally in a space. These paths have domain the unit interval and map into "small" neighborhoods of points in a space. Moreover ...

**4**

votes

**0**answers

40 views

### Intuition for universal quotient maps

The universal quotient maps are precisely the descent morphisms in the category of topological spaces. In some papers of Janelidze, Tholen, Sobral, and Reiterman (see for instance Reiterman-Tholen), ...