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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.

3
votes
1answer
201 views

Example of a Baire Class $1$ function $f$ satisfying $\omega\cdot n<\beta(f)\leq \omega\cdot (n+1)$ for some natural number $n\geq 1.$

Definitions: Let $X$ be a Polish space (separable completely metrizable topological space). A function $f:X\to\mathbb{R}$ is Baire Class $1$ if it is a pointwlise limit of a sequence of continuous ...
2
votes
1answer
151 views

Hausdorff quotient space with compact or finite inverse images

Let $X$ be a locally compact, second countable Hausdorff topological space and let $Y$ be a Hausdorff quotient of $X$. Let $q:X\to Y$ denote the quotient map. Then for $y\in Y$, $q^{-1}(y)$ is a ...
2
votes
1answer
140 views

Homeomorphic open sets and total disconnectedness

If $(X,\tau)$ is a $T_2$-space such that all non-empty open sets are homeomorphic (with the subspace topology) to $X$, is there for all $x,y\in X$ with $x\neq y$ a clopen (closed and open) set ...
8
votes
1answer
203 views

Examples of Baire Class $\xi+1$ but not $\xi$ functions for each countable ordinal $\xi.$

We say that $f:\mathbb{R}\to\mathbb{R}$ is of Baire Class $1$ if it is a pointwise limit of a sequence of continuous functions. One can generalize the definition above by taking pointwise limit of ...
4
votes
0answers
84 views

point-wise approximation of the identity in hereditary Lindelof spaces

Let $X$ be a topological vector space. Assume that there exists a sequence of finite range measurable functions $\phi_n:X\to X$ with $\lim\phi_n(x)=x$. Q. Can we concluded that $X$ is hereditery ...
4
votes
1answer
114 views

$n$-star-compactness

Let $X$ be a set. For $B\subseteq X$ and ${\cal H}\subseteq {\cal P}(X)$ we set $$\text{ST}^1(B,{\cal H}) = \{H\in {\cal H}: H\cap B\neq \emptyset\},$$ and $\text{st}^1(B,{\cal H}) = \bigcup \text{...
5
votes
2answers
253 views

An example of an open discontinuous function

Consider the following simple example of a function $f: \mathbb{R}\to\mathbb{R}$ which is open and discontinuous at all points. If $x\in\mathbb{R}$ is represented as something.$x_1x_2x_3\dots$ in the ...
9
votes
0answers
354 views

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$?

Is there a model of set theory in which $\mathfrak p< \mathfrak b < \mathfrak q$? Here $\mathfrak p$, $\mathfrak b$, $\mathfrak q$ are small uncountable cardinals: $\mathfrak p$ is the ...
5
votes
0answers
212 views

The second dual of $C(X)$ with the compact-open topology

Let $X$ be a compact Hausdorff space. Then $C(X)$ is a Banach algebra with the supremum norm and so is $A=C(X)^{**}$ under either Arens product. Moreover, it is easy to verify that $A\cong C(Z)$ for ...
14
votes
3answers
527 views

Version of Banach-Steinhaus theorem

I am wondering about the following version of the Banach-Steinhaus theorem. Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
2
votes
2answers
114 views

Thick refinements of covers

Let $(X,\tau)$ be a topological space, and let ${\cal U}$ be an open cover. We say that ${\cal U}$ is thick if for all $x\in X$ we have $$|\{V\in {\cal U}: x\in V\}| = |X|.$$ Is there a Hausdorff ...
3
votes
1answer
105 views

Is each closed subgroups of $\mathbb R^\omega$ isomorphic to a Tychonoff product of locally compact Abelian groups?

It is known that any closed linear subspace of $\mathbb R^\omega$ is topologically isomorphic to $\mathbb R^n$ for some $n\in\omega$. Problem 1. Is each closed subgroup of $\mathbb Z^\omega$ (or ...
6
votes
1answer
360 views

Homeomorphic open sets and homogeneity

If $(X,\tau)$ is a $T_2$-space such that all non-empty open sets are homeomorphic (with the subspace topology) to $X$, is $(X,\tau)$ necessarily homogeneous?
5
votes
3answers
199 views

Cogenerator of Categories of Topological Spaces Satisfying Some Separation Axiom

This question begins with a sort of mysterious comment at the bottom of this Wikipedia page on injective cogenerators. There, it is said, without citation or proof, that as a result of the Tietze ...
2
votes
0answers
94 views

description of dual space of space of Radon measure equipped with topology of weak convergence

Let $\mathcal{M}(\mathbb R)$ be the space of Radon measures, equipped with topology $\tau$ generated by the following "weak convergence": $$ \mu_n \rightarrow \mu \quad \text{iff} \quad \int f d\mu_n ...
3
votes
0answers
63 views

Which spaces are still Lindelöf after forcing with a Suslin tree?

Let $T$ be a Suslin tree and $f:T\to Y$ be continuous. ($T$ is endowed with the order topology.) Assume that the image of $T$ is contained in a Lindelöf subset of $Y$. Then, force with $T$. Which ...
3
votes
2answers
174 views

Signature of the manifold of the multiple fibrations over spheres

We can define the signature of a manifold in $4k$ dimensions. 1) If I understand correctly, the signature $\sigma$ of the manifold of the product space of spheres would always be zero: $$\sigma(S^...
0
votes
0answers
91 views

Question about upper semicontinuous functions

Let $p\in\mathbb N$ and $A$ a set. Denote by $[A]^{\leq p}$ the set of all subsets of $A$ with cardinality less than or equal to $p$. Does exist a compact Hausdorff space $K$ such that there is no ...
11
votes
3answers
2k views

Are uniformly continuous functions dense in all continuous functions?

Suppose that $X$ is a metric space. Is the family of all real-valued uniformly continuous functions on $X$ dense in the space of all continuous functions with respect to the topology of uniform ...
5
votes
1answer
295 views

Homeomorphism-fixing subsets

If $(X,\tau)$ is a topological space, we say $Y\subseteq X$ is homeomorphism-fixing if the only homeomorphism $\varphi:X\to X$ such that for all $y\in Y$ we have $\varphi(y)=y$ is the identity map. ...
5
votes
1answer
457 views

Nice partition of $\mathbb{R}$ into uncountably many uncountable sets

A recent issue of American Math. Monthly has a paper that partitions $\mathbb{R}$ into an arbitrary finite number of uncountable sets such that every real number is a condensation point of all the ...
4
votes
1answer
210 views

Approximation of the identity by finite range functions in topological vector spaces

Let $X$ be a topological vector space. Assume that there exists a sequence $\phi_n:X\to X$ of finite range measurable functions with $\lim\phi_n(x)=x$ for every $x\in X$. Can we concluded there exists ...
6
votes
5answers
306 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
196 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
113 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
301 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
129 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
295 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
140 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
104 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
74 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
108 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 ...
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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
432 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
77 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
167 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
75 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
137 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
260 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
333 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
78 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 ...
3
votes
1answer
196 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
200 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
512 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
197 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 ...