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Questions tagged [gn.general-topology]

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

6
votes
5answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
2answers
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
2answers
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
2answers
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
1answer
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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 ...