Questions tagged [gn.general-topology]
Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.
4,448
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On the homotopy type of $\mathbb{QP}^\infty$
It can be shown that the infinite-dimensional rational projective space $\mathbb{QP}^\infty$ is a connected, Hausdorff topological space. What can be said about its homotopy type (is it simply ...
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3
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0
answers
132
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Is there a normal space with a $G_\delta$ diagonal which is not submetrizable?
A space has a $G_\delta$-diagonal if its diagonal can be written as the intersection of countably many open subsets of the square. A space is submetrizable if it has a weaker metrizable topology. ...
- 4,343
10
votes
1
answer
304
views
Closed vector subspaces of large powers of R
By a large power of $\mathbb R$ is meant a topological vector space which is the product of infinitely many copies of the real line.
Is every closed subspace of such a TVS linearly homeomorphic to ...
- 101
5
votes
1
answer
154
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A regular first countable space of cellularity at most $2^\omega$
Let $X$ be a regular first countable space of cellularity at most $2^\omega$.
Is it true that the cardinality of $X$ is at most $2^\omega$?
A cellular family is a family of pairwise disjoint non-...
- 601
5
votes
1
answer
372
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Sufficient criteria for $X \subset \mathcal{H}$ to be a Lipschitz (or unif. cont.) retract of $\mathcal{H}$
I am interested in sufficient criteria which ensure that a subset $X$ of a Hilbert space $\mathcal{H}$ is a Lipschitz (or at least uniformly continuous) retract of $\mathcal{H}$.
Under which ...
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14
votes
2
answers
532
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Constructive proofs of existence in analysis using locales
There are several basic theorems in analysis asserting the existence of a point in some space such as the following results:
The intermediate value theorem: for every continuous function $f : [0,1] \...
- 4,410
3
votes
0
answers
153
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$G_\delta$-diagonal and productivity of the CCC
Is there a known example of a completely regular c.c.c. space with $G_\delta$-diagonal which is not productively c.c.c.?
The non-existence of such a space is consistent (for example, under $MA$ no ...
- 1,655
9
votes
1
answer
482
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Which topological manifolds do not correspond to strongly Hausdorff locales?
I'm toying with the idea of using locales as a way to define topological manifolds without beginning with points, largely for philosophical reasons.
In this context I think I want to redefine a ...
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11
votes
1
answer
2k
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What are compact objects in the category of topological spaces?
Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits.
On the other hand, ...
- 27.3k
1
vote
1
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107
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Hausdorff convergence of preimages of discrete-valued functions
Suppose $f_n$, $f:X\to K$ where $K$ is a finite set and $(X,d)$ is a metric space. Suppose also that $f_n(x)\to f(x)$ for all $x\in X$ (pointwise convergence). Finally, let $d_H$ be the Hausdorff ...
- 680
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78
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Is every weakly Lindelof Banach space a $D$-space?
An open neighbourhood assignment for a topological space $(X, \tau)$ is a map $U: X \to \tau$ such that $x \in U(x)$, for every $x \in X$. A space $X$ is called a $D$-space if for every open ...
- 4,343
3
votes
2
answers
342
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Paths in Cech closure spaces
Simply stated, I've been trying for a long time to either find in the literature, or derive myself, a notion of path in Cech closure spaces, that specialises to paths in a topological space, and to ...
- 415
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votes
1
answer
397
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Simplicial complex construction from given Betti numbers?
Is it possible given a set of Betti numbers to construct a (possibly set of) simplicial complex with the given Betti-described topology? I understand there can be an infinity of simplicial complexes ...
- 175
7
votes
1
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On the cardinality of ccc spaces with a $G_\delta$-diagonal
In a recent MO post it was noted that Uspenskij's old example of a Tychonoff ccc space with a $G_\delta$ diagonal and arbitrarily large cardinality is not normal. See:
How could I see quickly that ...
- 4,343
2
votes
1
answer
178
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Differentials in different topologies
I have read (In French)that the differential of a function depends on the topology and not the norm, the latter is rather easy to grasp, the first is hard for me to construct.
Norms being equivalent ...
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vote
1
answer
446
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Base of topology for metric-like space
Let $X$ be a nonempty set and $p:X\times X\rightarrow\mathbb{R}^+ $ be a function satisfying the following conditions for all $x,y,z\in X$: \begin{align} &1)\enspace p(x,y)=0\implies x=y \\ &2)...
3
votes
1
answer
197
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How could I see quickly that this space is not normal?
Recently, I read a paper in which the author construct a space $X$ which is dense in a $\sigma$-product $S$ of closed unit intervals. The space $X$ is CCC (denotes countable chain condition); it is ...
- 601
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0
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541
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Small cardinals related to topological convergence
Recall that a topological space is called sequential if a set is closed if and only if it contains all limits of convergent sequences lying inside of it. A space $X$ is called Frechet if for every non-...
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1
vote
1
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132
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Let $X$ be a Lindelof, perfectly normal, $\sigma$-space. Must $X$ be separable?
A space $X$ is a $\sigma$-space if $X$ has a $\sigma$-discrete network.
Let $X$ be a Lindelof, perfectly normal, $\sigma$-space.
Must $X$ be separable?
Thanks very much.
- 601
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0
answers
427
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Motivation for studying group of homeomorphisms of topological spaces [closed]
Currently I am reading a paper titled "On the Group of Homeomorphisms of an Arc" by N.J Fine and G.E. Schweigert, that was published by Annals of Mathematics in 1955. This paper talks about the group ...
- 428
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votes
1
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521
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Could we always find a line to intersect transversally with a given compact manifold?
This problem may be an embarrassing one, but I could not prove it even for the $1$ dimensional case. Here is the problem:
Question 1. $M$ is a compact $n$-dimensional smooth manifold in $R^{n+1}$. ...
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13
votes
1
answer
644
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Frankl's conjecture restricted to finite topological spaces
A finite topological space is a finite family of finite sets that is closed under both union and intersection.
Frankl's conjecture states that for any finite union-closed family of finite sets, ...
- 2,539
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0
answers
220
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When can we force two frames to be homeomorphic?
Recall that if $M,N$ are two structures of the same type, then
$M$ is $\mathcal{L}_{\infty,\omega}$ elementarily equivalent to $N$ precisely when $M$ and $N$ are isomorphic in some forcing extension. ...
- 27.8k
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votes
1
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128
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Is there a $\sigma$-metacompact space which is not metacompact?
Recall that a space $X$ is metaLindelof if every open cover of
$X$ has a point-countable open refinement.
A space $X$ is metacompact if every open cover of
$X$ has a point-finite open refinement....
- 601
5
votes
2
answers
198
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Connected $T_2$-space with $\text{Cont}(X,X)$ not dense in $X^X$
Disclaimer: Feel free to downvote or vote to close, if this is again trivial (I seem to have a bad day today; I promise that if this is again a bummer question, I will wait $\geq 1$ day before asking ...
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2
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475
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Is $\text{Cont}(\mathbb{R},\mathbb{R})$ dense in $\mathbb{R}^\mathbb{R}$? [closed]
Let $\text{Cont}(\mathbb{R},\mathbb{R})$ denote the set of continuous self-maps of $\mathbb{R}$ and let $\mathbb{R}^\mathbb{R}$ denote the set of all self-maps of $\mathbb{R}$, endowed with the ...
- 45.5k
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Are continuous self-maps of the Golomb space $\mathbb G$ dense in the space of all self-maps of $\mathbb G$?
The Golomb space $\mathbb G$ is the set $\mathbb N$ of positive integers endowed with the topology generated by the base consisting of arithmetic sequences $a+b\mathbb N_0:=\{a+bn:n\ge 0\}$ with $a,b$ ...
- 40.9k
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462
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If non-empty player has a winning strategy in Banach-Mazur game BM(X), then it also has in BM(Y)?
Let $f:X\rightarrow Y$ be a continuous, open, surjection function and second player (non-empty) has a winning strategy (not important which one, say for simplicity stationery st.) in $BM(X)$. Then can ...
- 31
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votes
1
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Countably infinite connected Hausdorff space with the fixed point property
Is there an infinite, countable connected $T_2$-space $(X,\tau)$ such that $(X,\tau)$ has the fixed point property? (This means that for every continuous map $f:X\to X$ there is $x\in X$ such that $f(...
- 45.5k
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vote
1
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269
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Cantor set onto connected set?
Let $X$ be a Hausdorff space such that the irrationals $\mathbb P$ (in their usual topology) form a dense subspace of $X$.
Let $C$ be the Cantor set. The set of "non-endpoints" of $C$ is ...
- 955
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votes
1
answer
478
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Is Bing's countable connected space topologically homogeneous?
In this paper R.H. Bing has constructed his famous example of a countable connected Hausdorff space.
The Bing space $\mathbb B$ is the rational half-plane $\{(x,y)\in\mathbb Q\times \mathbb Q:y\ge 0\...
- 40.9k
6
votes
1
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250
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Is the Mackey topology $\tau(l^{\infty},l^{1})$ strongly Lindelöf?
Let $l^{\infty}$ (respectively, $l^{1}$) be the space of bounded
(respectively, absolutely summable) real sequences. I need to find out if
$l^{\infty}$ equipped with the Mackey topology $\tau(l^{\...
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1
answer
450
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A criterion for second countability
Let $(X,\tau)$ be a topological space.
Assume for any arbitrary topological base $\mathcal{E}$ of $\tau$ we have that: the Borel sigma algebras coming form $\mathcal{E}$ and $\tau$ are the same. ...
- 3,992
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votes
0
answers
138
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Disjoint covering number of an ideal
Let $\mathcal I$ be a $\sigma$-ideal with Borel base on an uncountable Polish space $X=\bigcup\mathcal I$.
Let $\mathrm{cov}(\mathcal I)$ (resp. $\mathrm{cov}_\sqcup(\mathcal I)$) be the smallest ...
- 40.9k
18
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8
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Concepts in topology successfully transferred to graph theory and combinatorics with non-trivial applications?
What are some of the difficult concepts in topology that have been transferred to graph theory and combinatorics where a certain new application has been found.
A good example is Lovász's proof of ...
3
votes
1
answer
142
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The Wallman and interval topologies on non-principal ultrafilters with the Rudin-Keisler preorder
If $(P,\leq)$ is a pre-odered set (that is, $\leq$ is a reflexive and transitive relation) and $x\in P$, we set $(\uparrow_{\leq} x) = \{p\in P: p\geq x\}$ and $(\downarrow_{\leq} x) = \{p\in P: p\leq ...
- 45.5k
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0
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343
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Cellular-Lindelöf: a common generalization of the Lindelöf property and the CCC
All spaces are assumed to be Hausdorff. Recall that a cellular family in the space $X$ is a family of pairwise disjoint non-empty open subspaces of $X$. The cellularity of $X$ ($c(X)$) is defined as ...
- 4,343
2
votes
1
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102
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Connected $T_2$ space with essentially no connected subspaces
Is there a connected $T_2$ space $(X,\tau)$ with more than one point, such that the singletons and $X$ are the only connected subspaces of $X$?
- 45.5k
3
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1
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128
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Is there a metacompact, normal, CCC space which is not Lindelof
I am looking for a space as in the title, i.e.,
Is there a metacompact, normal, CCC space which is not Lindelof?
A space is ccc iff any family of pairwise disjoint open sets is at most countable.
...
- 601
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votes
1
answer
156
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Continuous self-maps in the Golomb space that are neither increasing nor decreasing
Let $\mathbb{N}$ denote the set of the positive integers. The Golomb space is a space ${\bf G} =(\mathbb{N},\tau)$ where a basis of $\tau$ is generated by
$$\big\{\{a+bn: n\in \mathbb{N}\cup\{0\}\}: a,...
- 45.5k
6
votes
1
answer
230
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Does $[0,1]\cap \mathbb{Q}$ have a connected $T_2$ quotient?
Is there an equivalence relation $R$ on $[0,1]\cap \mathbb{Q}$ such that $([0,1]\cap \mathbb{Q})/R$ is connected, Hausdorff, and has more than $1$ point?
- 45.5k
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votes
2
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Is there a "universal" connected compact metric space?
Fact 1. The Cantor set $K$ is "universal" among nonempty compact metric spaces in the following sense: given any nonempty compact metric space $X$, there exists a continuous surjection $f\colon K \to ...
- 21.5k
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votes
2
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2k
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Are the Sierpiński cardinal $\acute{\mathfrak n}$ and its measure modification $\acute{\mathfrak m}$ equal to some known small uncountable cardinals?
This question was motivated by an answer to this question of Dominic van der Zypen.
It relates to the following classical theorem of Sierpiński.
Theorem (Sierpiński, 1921). For any countable partition ...
- 40.9k
19
votes
3
answers
1k
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"Anti" fixed point property
Let $(X,\tau)$ be a topological space. If $f:X\to X$ is continuous, we say $x\in X$ is a fixed point if $f(x) = x$.
The space $(X,\tau)$ is said to have the anti fixed point property (AFPP) if the ...
- 45.5k
3
votes
1
answer
94
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Is there a calibre $\aleph_1$ Moore space which is not separable
A topological space has calibre $\aleph_1$ if for every uncountable sequence $\langle U_\alpha\mid\alpha\lt\aleph_1\rangle$ of nonempty open sets $U_\alpha\subset X$, there is an uncountable subfamily ...
- 601
1
vote
0
answers
195
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free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$
I want to construct free $S^1$ action on $\mathbb{R}P^n$ and $\mathbb{C}P^n$.
For $n=2m-1$, consider $S^n ⊂ C^m$. Then $S^1$ freely act on $S^n$ by $(ξ, (z_1 , z _2 , · · · , z _m )) → (ξz_1 , ξz_2 ,...
- 395
3
votes
0
answers
55
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Name for a special kind of neighborhood assignment or for the existence thereof
Lets say temporarily that a topological space $(X,\tau)$ is weird if there is a function $\varphi:X \to \tau$ such that for all $x \in X$:
$x\in\varphi(x)$,
$\{y\in X: x \in \varphi(y)\}$ is finite.
...
- 11.5k
1
vote
2
answers
174
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Non-homogeneous space $X$ such that $X\cong X\setminus \{x\}$ for all $x\in X$
What is an example of a topological space $(X,\tau)$ with the properties that
$X\cong X\setminus \{x\}$ for all $x\in X$, and
$(X,\tau)$ is not topologically homogeneous
?
- 45.5k
8
votes
0
answers
128
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Local vs global homogeneity of topological spaces
I am interested in the relation between global and local homogeneity of topological spaces. On one extreme we have rigid spaces, i.e., topological spaces with trivial homeomorphism group.
Question. ...
- 40.9k
1
vote
1
answer
165
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Miscenko example of linearly Lindelof non Lindelof is not normal
In the paper of Norman Howes "A note on transfinite sequences" is mentioned that Miscenko space
$M = \{f \in \prod_{n \in \omega \smallsetminus \{0\}} \aleph_n+1 | \space \exists k \space \space \...
- 11