**3**

votes

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**3**answers

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

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**2**answers

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**0**answers

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

**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

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 $C(X)$...