# Questions tagged [gn.general-topology]

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

**6**

votes

**5**answers

310 views

### If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we have $|X| =|\tau|$?

If $(X,\tau)$ has more than $1$ point and is $T_2$ and connected, do we necessarily have $|X| =|\tau|$?

**1**

vote

**0**answers

113 views

### A special topological property

Let us say a topological space $X$ is a countable union of second countable spaces if there exists a sequence of subsets $\{X_n\}$ of $X$ with $X=\cup X_n$ such that the relative topology on $X_n$'s ...

**4**

votes

**1**answer

210 views

### Is each Swiatkowski function with closed graph continuous?

A function $f:\mathbb R\to\mathbb R$ is called Świątkowski if for any connected subset $C\subset \mathbb R$ and points $a,b\in C$ with $f(a)<f(b)$ there exists a continuity point $x\in C\setminus\{...

**1**

vote

**1**answer

126 views

### $S_M$ is not always homeomorphic to the 1-sphere of $F$

Let $F$ be a infinite-dimensional complex Hilbert space, with inner product $\langle\cdot\;| \;\cdot\rangle$, the norm $\|\cdot\|$, the 1-sphere $S(0,1)=\{x\in F;\;\|x\|=1\}$ and let $\mathcal{B}(F)$ ...

**6**

votes

**1**answer

116 views

### Example similar to the Griffiths twin cone but with fundamental group that allows surjection onto $\mathbb Z$

The Griffiths twin cone is an example of a wedge sum of two contractible spaces being non-contractible. Namely, it is the wedge sum $\mathbb G=C\mathbb H\vee_p C\mathbb H$ of two coni over the ...

**10**

votes

**1**answer

303 views

### Are all compact subsets of Banach spaces small in a measure-theoretic sense?

Definition. A subset $K$ of a topological group $X$ is called measure-continuous if there exists a $\sigma$-additive Borel probability measure $\mu$ on $X$ such that for every compact subset $C\subset ...

**3**

votes

**1**answer

130 views

### Is there a connected $T_2$-topology on $\mathbb{Q}$ that is coarser than the Euclidean one?

Let $\mathbb{Q}$ be the rationals, and let $\tau$ be the Euclidean topology on $\mathbb{Q}$. Is there a topology $\tau' \subseteq \tau$ such that $(\mathbb{Q},\tau')$ is connected and $T_2$?

**8**

votes

**2**answers

301 views

### Approximation of the identity by simple functions

Let $X$ be a topological space. Assume that there exists a sequence of simple functions $\phi_n:X\to X$ (finite range and measurable) with $\lim\phi_n(x)=x$.
Can we concluded $X$ may be written by a ...

**3**

votes

**1**answer

142 views

### Are second-countable subsets of topological vector spaces metrizable?

Let $X$ be a topological vector space of size $\mathfrak{c}$. Assume that there exists a countable union $X=\cup X_n$ such that all subsets $X_n$'s are relatively second countable.
Q. Does there ...

**3**

votes

**1**answer

106 views

### Approximation on separable topological space with size $\mathfrak{c}$

Let $X$ be a separable topological space of size $\mathfrak{c}$. By a simple function $\phi:X\to X$, we mean a finite range valued measurable function.
Q. Is it possible to find a sequence of ...

**3**

votes

**1**answer

77 views

### Connected $T_2$-space such that the open sets are closed under countable intersection

This question has a trivial starting point: If the open sets of $\mathbb{R}$ were closed under countable intersection, the Euclidean topology would be discrete because for all $x\in\mathbb{R}$ we have ...

**4**

votes

**0**answers

109 views

### A point concerning Fremlin's example on Borel sets in non-separable Banach spaces

Let $E$ be a Banach space. Let us consider the following three sigma algebras on $E$.
$~~~~\mathcal{B}$= The $\sigma$-algebra coming from the norm topology.
$~~\mathcal{M}$= The sigma algebra ...

**1**

vote

**0**answers

130 views

### Comparing two $\sigma$-algebras

Let $X$ be a set. We denote $P(X)$ by the family of all subsets of $X$. We also denote $P(X)\otimes_{\sigma}P(X)$ by the $\sigma$-algebra generated by $\{A\times B: A,B \subseteq X\}$.
Q. For which ...

**13**

votes

**1**answer

441 views

### $T_2$-spaces where all non-empty open sets are homeomorphic

We say that a $T_2$-space $(X,\tau)$ has homeomorphic open sets if every non-empty open set $U\subseteq X$ endowed with the subspace topology is homeomorphic to $(X,\tau)$.
The rationals with the ...

**4**

votes

**0**answers

78 views

### A compact set in the plane with small sum-set and large projections

Problem. Let $K$ be a compact subset of the plane such that the projection of $K$ on each line has non-empty interior in the line. Has $K+K$ or $K-K$ non-empty interior in the plane?
Remark. The ...

**2**

votes

**1**answer

136 views

### Boundedness of Dirac deltas

Suppose that $X$ is a metric space and let $C_k(X)$ denote the space of real functions on $X$ with the topology of uniform convergence on compact sets. Then $C_k(X)$ is a topological vector space. Let ...

**1**

vote

**1**answer

168 views

### Connected Hausdorff spaces with different cardinalities of open sets

Given an infinite cardinal $\kappa$, is there a connected Hausdorff space $(X,\tau)$ with $|X|=\kappa$, and for every infinite cardinal $\lambda \leq \kappa$ there is an open set $U\in \tau$ with $|U| ...

**0**

votes

**1**answer

77 views

### Is $C_b(Q,E)$ linearly isometric isomorphic to $C(\beta Q,E)$ where $\beta Q$ is the Stone-Cech compactification of $Q?$

Let $Q$ be a locally compact Hausdorff space and $E$ be a Banach space.
Let $C(Q)$ be the collection of all real-valued continuous functions on $Q$ while $C_b(Q,E)$ be the collection of all $E$-...

**9**

votes

**1**answer

287 views

### Comparing two $\sigma$-algebras on $B(\ell^1)$

Let us consider $B(\ell^1)$, bounded linear operators on $\ell^1$. We recall the weak operator topology, denoted by $w$, on $B(\ell^1)$ is determined as follow
$$w-\lim T_i=T \Longleftrightarrow \...

**4**

votes

**1**answer

146 views

### Paracompactness of Quotient by Group Action

Suppose $X$ is a metric space with a free group action by a topological group $G$, which is also a metric space, such that $\pi\colon X \to X/G$ is a fiber bundle.
Does the quotient inherit the ...

**3**

votes

**2**answers

264 views

### What are the components of the Stone-Cech Remainder?

Suppose $X = \displaystyle\bigsqcup_{i \in I} X_i$ is the disjoint union of infinitely many continua. The components of the Stone-Cech remainder $X^*$ can be described as follows. Treat $I$ as a ...

**1**

vote

**0**answers

102 views

### A section over an orbit space

Let $G$ be a compact second countable Hausdorff group, and let $X=G/H$ be a homogeneous space with $H\subset G$ a closed subgroup. Let further $K\subset G$ be another closed subgroup.
Questions:
...

**5**

votes

**2**answers

334 views

### Do a Hausdorff space and its associated completely regular space have the same Borel subsets?

Let $(X,T)$ be a Hausdorff topological space. Let $C_b(X)$ be its algebra of continuous bounded functions. Let $T'$ be the initial topology on $X$ given by $C_b(X)$. It is known that $T=T'$ if and ...

**2**

votes

**2**answers

80 views

### Does any subset of $\beta\omega$ of cardinality $\mathfrak{c}$ have a weak P-point in its closure?

I believe that anyone who can answer this question knows the terminology, but $\beta\omega$ is the Čech-Stone compactification of the integers and a point $p$ is a weak P-point in a space $X$ iff it ...

**4**

votes

**1**answer

203 views

### Brouwer fixed-point for open ball and bijective uniformly continuous function?

Let $B^-$ be the open unit ball in $\mathbb{R}^n$ for $n\in\mathbb{N}$, and let $f$ be a uniformly continuous function from $B^-$ to itself, with a uniformly continuous inverse $f^{-1}$.
Under these ...

**1**

vote

**2**answers

221 views

### Understanding reduced suspension of $S^1$ [closed]

I know this is just $S^2$. To see it, I use the CW structure of $S^1$ x $S^1$ , consisting of one 0-cell, two 1-cells and a 2-cell. Then since the reduced suspension is the cartesian product ...

**10**

votes

**0**answers

518 views

### A standard name for a function satisfying the intermediate value theorem?

Do you know any (standard) name for a function $f:\mathbb R\to\mathbb R$ having the following weak intermediate value property:
$(*)$ for any connected subset $C\subset \mathbb R$ and points $a,b\...

**4**

votes

**1**answer

200 views

### Separable Lindelöf locally convex spaces that are not second-countable

A Lindelöf space is a topological space in which every open cover has a countable subcover.
Does there exists a Lindelöf locally convex space which is not second countable?
I am also looking for a ...

**3**

votes

**1**answer

132 views

### Distance for $GL_n(\mathbb{R})/GL_n(\mathbb{Z})$

One can define the convergence of a sequence $(\Lambda_k)_k$ of full rank lattices as folow : $(\Lambda_k)\underset{k\rightarrow +\infty}{\longrightarrow} \Lambda \iff \forall k\in \mathbb{N} ,\exists ...

**5**

votes

**1**answer

156 views

### Direct limit of Cantor sets

Let $C$ be the Cantor set, and $\omega$ the discrete space of integers $\{0,1,2,...\}$.
My conjecture:
(1) For each $n<\omega$ let $f_n:C\to C$ be a continuous function (possibly not onto). Let ...

**0**

votes

**1**answer

68 views

### Existence of certain subsemigroups of $C(K, K)$ for compact Hausdorff spaces $K$

Let $K$ be a compact Hausdorff space. I'm wondering: Does there always exist a subset $J \subseteq C(K, K)$ such that:
$J$ is closed under composition,
there is an element $f \in C(K)$ such that the ...

**5**

votes

**1**answer

309 views

### A question on Mobius strip and Jordan curve

If $A\subset \Bbb R^2$ then is the following statement true?
$\{(x,y)\in {(A\times A)/ \sim}\,\,\,|\,\, (x,y)\sim(y,x)\}\simeq \text{Mobius strip}\quad\iff\quad A\,\,\, \text{is a Jordan curve}.$...

**8**

votes

**2**answers

189 views

### Spaces without maximal homogeneous subspaces

A homogeneous space $(X,\tau)$ is a topological space such that for all $x,y\in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x)=y$. As a previous question implies, the union of an ...

**3**

votes

**0**answers

58 views

### On the compactification of partial semigroups

We begin by introducing some relevant definitions.
Definition: A $\textit{partial semigroup}$ is a pair $(S,.)$ where $.$ maps a subset of $S \times S$ to $S$ and for all $a,b,c \in S, (a.b).c=a.(b.c)...

**1**

vote

**1**answer

111 views

### Is a local diffeomorphism with nice boundary values a diffeomorphism?

Let $f:\mathbb{D}=\{z\in\mathbb{C}\mid |z|<1\}\rightarrow\mathbb{C}$ be a local diffeomorphism (i.e. an immersion) from an open disk in the plane to the plane.
The only situation I can image ...

**6**

votes

**1**answer

85 views

### Name for $\omega_1$-DCC / Noetherian condition?

I recently asked (and then answered) this question:
https://math.stackexchange.com/questions/2756777/decreasing-sequence-of-closed-sets-in-a-separable-metric-space.
In a separable metric space ...

**1**

vote

**0**answers

42 views

### A criterion for metrizable topological spaces [duplicate]

Let $X$ be a topological space. True or false?
$X$ is metrizable if and only if it contains a sequence of metrizable spaces $\{X_n\}$ with $X=\bigcup X_n$!

**9**

votes

**2**answers

167 views

### Minimal refinements of open covers of $T_2$-spaces

Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if
$\bigcup {\cal U} = X$, and
$X\notin {\cal U}$.
${\cal U}$ is minimal if for all $U_0\in {\cal U}$ we have ...

**3**

votes

**1**answer

147 views

### Arcwise-connectedness generalized to higher connectivity?

This is a crosspost from stackexchange. I'm not completely sure whether the question below is research-level, but I have not yet found an obvious answer, and what I have found thus far suggests that ...

**5**

votes

**1**answer

226 views

### Is each compact metric space a subset of a compact absolute 1-Lipschitz retract?

A metric space $X$ is called an absolute $L$-Lipschitz retract if for any metric space $Y$ containing $X$ there exists a Lipschitz retraction $r:Y\to X$ with Lipschitz constant $Lip(r)\le L$.
...

**1**

vote

**0**answers

74 views

### When is a nested sequence of closed sets a colimit?

Let $X$ denote a topological space and $X_0\subset X_1\subset \ldots\subset X$ a nested sequence of closed subsets of $X$ such that $$ \bigcup_i X_i =X$$
It is easy to see that in the general case $X$...

**2**

votes

**1**answer

88 views

### Measurability of the product on particular topological vector spaces

Let $X$ be a topological vector space. Let us say that $X$ has property P if there exists a sequence of closed subsets $\{X_n\}$ such that
1- $X=\bigcup X_n$
2- The relative topology is both ...

**8**

votes

**0**answers

92 views

### Connected component optimization

For an open set $A\subset[0,1]^d$, denote the connected components of $A$ by $cc(A)$. Given a smooth symmetric function $f\colon[-1,1]^d\to\mathbb R$ with $f(0)>0$, I am interested in the ...

**2**

votes

**2**answers

262 views

### covering theory with compact open topology

In the following all spaces $C^0(X,Y)$ are spaces of base point preserving maps with the compact-open topology.Furthermore all spaces I consider in the following are locally pathwise connected.
Under ...

**9**

votes

**1**answer

303 views

### Can every dense subset be partitioned into two dense subsets?

The question was very popular over on MSE but seems to have left everyone speechless. Maybe someone here can help?
Definition: Suppose $X$ is a compact connected Hausdorff space and $D \subset X$ ...

**8**

votes

**1**answer

305 views

### Why $S$ cannot be homeomorphic to the $1$-sphere of $\ell^2$?

Consider the $\ell^2$ complex Hilbert space.
Let $m\in \mathbb{N}^*$ be a fixed number, and set
$$
S=\left\{ x=(x_n)_n\subset \ell^2\ :\ \sum_{n=1}^m \frac{|x_n|^2}{n^2}=1\right\}.$$
I want to ...

**4**

votes

**0**answers

188 views

### Set of subsequences with the same ultrafilter limit of the original sequence

Let $\mathscr{U}$ be a free ultrafilter on the positive integers $\mathbf{N}$ and fix $U \in \mathscr{U}$ such that $U$ is not cofinite (thanks J.D.Hamkins for the correction.)
Consider the natural ...

**2**

votes

**1**answer

246 views

### Embedding into $C\times [0,1]$

Every totally disconnected separable metric space of dimension $n$ homeomorphically embeds into $C\times \mathbb R ^n$.
Is something like this known? $X$ is totally disconnected means that every ...

**0**

votes

**0**answers

29 views

### A specific weak analog of the Baire category theorem for a 'continuously indexed' family of sets

(This is a refinement/repost of a question I asked on Stack Exchange.)
Suppose that $C\subseteq [0,1]$ is a Cantor set (i.e. a totally disconnected closed perfect set) and $F\subseteq C\times [0,1]$ ...

**5**

votes

**1**answer

351 views

### One point compactification of $(\mathbb{C}^{\ast})^n$

I would like to know if there is a closed form formula for the homotopy type of $\widehat{(\mathbb{C^{\ast}})^n}$? For example, it is not difficult to see that $\widehat{\mathbb{C^{\ast}}}$ has the ...