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

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3
votes
1answer
54 views

Is every regular Borel outer measure topologically additive?

If $m$ is a regular Borel outer measure is it true that $m$ is topologically additive? If so what is a proof or a counterexample? Definitions: Topologically Additive: $X$ is a topological space, $m$ ...
0
votes
0answers
26 views

Jordan curve in $C^2$ [migrated]

Can we find a Jordan curve $\gamma$ in $\mathbf{C}^2$ of class $C^1$ such that the projection to the first coordinate plane divides the plane into infinite components of connectivity.
0
votes
0answers
86 views

Join of $G$-CW-Complexes

I want to understand the CW-structure on the join of $G$- CW complexes for my master's thesis. Let $G$ be a discrete group and $X$ and $Y$ $G$-CW-complexes. Furtheremore, let $X*Y$ denote the join $$[...
1
vote
1answer
94 views

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 \...
-2
votes
0answers
75 views

If we find compactification of dense subset of a topological spaces, then what can we say about compactification of original space? [closed]

Let $(X,T)$ be a topological space and $F$ a dense subset of $X$. Suppose that we have compactification of $(F,T)$, $(f,\beta F)$. Does $(X,T)$ possess a compactification?
-4
votes
0answers
65 views

If there exist compact dense subset of a topological space, then can we say about the compactiness of topological space? [closed]

Let (X,T) be a topological space and there is A subset of X and is compact then (X,T) is compact? If not then what be the counter example for this statement?
-2
votes
0answers
39 views

metrizable topology and furstenberg's topology [closed]

Let $X$ be a metrizable topological space. Are there methods for constructing a metric which induces the topology of $X$? And,pls, Is anyone aware of any problem related to opened Furstenberg's ...
0
votes
0answers
65 views

If $X_0$ is very dense in $X$ and $A \cap X$ is closed, then what is $A$?

Let $X$ be a scheme and let $X_0$ be a very dense subset (e.g., $X$ a finite type scheme over a field and $X_0$ the closed points). If $A$ is an arbitrary subset of $X$ such that $X_0 \cap A$ is ...
1
vote
0answers
76 views

Induced structure of topological group [closed]

If we consider a closed Jordan curve $\mathcal{C}$, I know that it's homeomorphic to the circle $S^1$. Now I take an homeomorphism $\phi:S^1\longrightarrow\mathcal{C}$ and this homeomorphism induces a ...
-5
votes
0answers
49 views

Is a continuous two variables function also continuous with respect to each variable? [closed]

I have a simple question, let $f:X\times Y\rightarrow Z$ be a map with two variables, and $X,Y,Z$ are topological spaces, I want to know if $f$ is continuous, then how about $f_{x_{0}}:Y\rightarrow Z$ ...
1
vote
0answers
62 views

Homotopy invariant deletions of open faces of simplicial complexes

Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball ...
6
votes
3answers
330 views

Lifting symmetries to the universal cover

If $X$ is a connected topological space with universal cover $p: \tilde{X} \to X$, I believe any homeomorphism $f : X \to X$ can be 'lifted' to a homeomorphism $\tilde{f} : \tilde{X} \to \tilde{X}$. ...
0
votes
0answers
23 views

Is the inverse of Minkowski's question mark function continuous on the dyadic fractions? [migrated]

I'm looking for a continuous function from the dyadic fractions between 0 and 1 to the rational numbers between 0 and 1. The inverse of Minkowski's question mark (also known as Conway's box function) ...
4
votes
0answers
81 views

Homeomorphism between evenly spaced integer topology and the rationals

The evenly spaced integer topology is countable, metrizable, and has no isolated points, and hence is homeomorphic to the rationals with the order topology. But what is an explicit construction for ...
2
votes
0answers
45 views

Is each Lindelof closed $\bar G_\delta$-set of a Tychonoff space functionally closed?

A subset $F$ if a topological space $X$ is called functionally closed if $F=f^{-1}(0)$ for some continuous map $f:X\to[0,1]$. It is clear that each functionally closed set $F$ in $X$ is a closed $G_\...
3
votes
1answer
126 views

Does the Lebesgue measure induce a finitely additive measure on the Boolean algebra of regular open subsets of (0,1)?

Let $(0,1)$ the unit interval. An open subset $\mathcal{R}\subseteq(0,1)$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. Unfortunately, ...
1
vote
1answer
145 views

Finitely additive measures on Boolean algebras of regular open subsets: Is there a relationship with Borel measures? A theory of integration?

Let $\mathcal{X}$ be a topological space. An open subset $\mathcal{R}\subseteq\mathcal{X}$ is regular if it is the interior of its own closure. The intersection of two regular open sets is regular. ...
1
vote
0answers
56 views

Inverse limits of the interval with a single bonding map below the identity

My question is as follows. QUESTION. Is there a topological description of the class of arc-like continua that arise as inverse limits of $[0,1]$ with a single continuous surjective bonding map $f\...
0
votes
0answers
70 views

Weak Topology and Domain Theory: Which topology on the function domain restricts to the weak topology on C([0,1])?

Let $\mathbb{IR}$ be the interval domain over the set $\mathbb{R}$ of real numbers, defined by: $$\mathbb{IR} := \{ [a,b] \mid a, b \in \mathbb{R}, a \leq b\} \cup \{ \mathbb{R}\},$$ and ordered by ...
9
votes
1answer
238 views

Cofinal monotone maps from $\omega^\omega$ to $\kappa^\kappa$

Given a cardinal $\kappa$ consider the set $\kappa^\kappa$ of all functions from $\kappa$ to $\kappa$, endowed with the partial order $f\le g$ iff $f(\alpha)\le g(\alpha)$ for all $\alpha\in\kappa$. ...
5
votes
0answers
95 views

Preservation of Baumgartner's I-ultrafilters under various forcings

For $I\subset \mathcal{P}(2^\omega)$, an ultrafilter $U$ on $\omega$ is said to be an I-ultrafilter if for all $f:\omega \to 2^\omega$, there exists $A\in U$ such that $f''[A]\in I$ [Baumgartner]. In ...
6
votes
1answer
195 views

Extending the topology on a set to the group/vector space it generates

The multiplicative group $\Bbb Q^+$ can be viewed as a $\Bbb Z$-module. To see this, note that any rational can be decomposed into the form $2^{n_2} \cdot 3^{n_3} \cdot 5^{n_5} \cdot ...$ The tuple ...
5
votes
0answers
110 views

Has a continuous map from $\kappa^\omega$ to $[0,1]^\omega$ a non-scattered fiber?

Question. Let $\kappa>\mathfrak c$ be a cardinal endowed with the discrete topology and $f:\kappa^\omega\to[0,1]^\omega$ be a continuous map. Is there a point $y\in[0,1]^\omega$ whose preimage $f^{-...
5
votes
1answer
160 views

Bernstein sets of large cardinality

A metrizable space $X$ will be called a generalized Bernstein set if every closed completely metrizable subspace $C$ of $X$ has cardinality $|C|<|X|$. It is well-known that the real line contains ...
8
votes
0answers
179 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\...
1
vote
1answer
88 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 \...
1
vote
2answers
98 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 ...
6
votes
1answer
291 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
99 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 ...
3
votes
1answer
185 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 ...
3
votes
0answers
149 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
2answers
117 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 ...
13
votes
2answers
436 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
69 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
451 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 ...
2
votes
2answers
79 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
110 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
89 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
180 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 ...
2
votes
1answer
91 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
77 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
180 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
71 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
66 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
188 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
263 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
295 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
49 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
501 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 $C(X)$...