Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

learn more… | top users | synonyms (2)

0
votes
0answers
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
0answers
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
0answers
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
1answer
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
2answers
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
0answers
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
2answers
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
0answers
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
2answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
0answers
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
0answers
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
2answers
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
0answers
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
0answers
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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), ...