Questions tagged [gn.general-topology]

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

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27 votes
3 answers
1k views

Possible categorical reformulation for the usual definition of compactness

Let $X$ be a compact topological space, $f_i:Y_i\to X$ a family of continuous maps such that the topology on $X$ is final for it (i.e., $U\subset X$ is open iff $f_i^{-1}(U)$ is open for each $i$, for ...
5 votes
0 answers
110 views

Free topological groups as smooth infinite-dimensional manifolds

It is known that the Graev free topological group $F(I)$ of the segment $I=[0,1]$ is homeomorphic to $\mathbb R^\infty=\lim\limits_{\longrightarrow}\mathbb R^n$. Is it possible to make an ...
11 votes
0 answers
319 views

Does any real function have a Lipschitzian restriction on $D$?

Does any real function have a Lipschitzian restriction on $D$, where $D$ is an infinite subset of $\Bbb R$ with an accumulation point?
6 votes
0 answers
162 views

J. F. Adams Proof of Cellular Approximation Theorem

In Ronald Brown's discussion of the proof of The Cellular Approximation Theorem in Topology and Groupoids Sec. 7.6 he writes that, "the elegant formulation of the proof is due to J. F. Adams." Does ...
3 votes
1 answer
165 views

Does a homeomorphism of open cones restrict to a quotient map of the bases?

Write $CX$ for the (pointed, or reduced) cone on $X$, and $C^\circ X$ for the open cone inside of it. Let's say a cone map is a map $g:CX\to CY$ such that $g(C^\circ X) \subseteq C^\circ Y$ and $g(X) ...
71 votes
1 answer
2k views

Dualizing the notion of topological space

$\require{AMScd}$ Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements ...
3 votes
1 answer
92 views

$T_2$-space with a matching equalling the density number

Given a topological space $(X,\tau)$, we say that a matching is a collection of non-empty open sets that are pairwise disjoint. Given an infinite cardinal $\kappa$, is there a $T_2$-space with $|X|\...
4 votes
0 answers
88 views

Pairs not at maximal distance in compact set

Does there exist a compact subset $K$ of $\mathbb{R}^2$ and a point $p$ in $\mathbb{R}^2$ with the following property? Let $x$ and $y$ be a pair of points in $K$ such that \begin{equation} |x-y| <...
1 vote
1 answer
166 views

Ordering a subset of the clopens of a Stone space

Let $P$ be a countably infinite set of propositional variables and $\mathcal{L}_P$ be the propositional language generated from $P$ and the usual connectives $\wedge$, $\neg$, $\vee$. The set $\...
4 votes
1 answer
261 views

Nowhere compact subsets of the plane

Suppose $X\subseteq \mathbb R^2$ is nowhere compact ($X$ has no compact neighborhood) and non-empty. Can $X$ be densely embedded into the plane? In other words, is there a dense set $X'\subseteq ...
6 votes
1 answer
534 views

Extension of Baire's Theorem

Let $X$ be a topological space, $\kappa$ be a cardinal number, such that there exists a dense subset $A\subseteq X$ of cardinality $\kappa$ but there does not exist a dense subset $A'\subseteq X$ of ...
8 votes
1 answer
616 views

Space filling curve whose all level sets are finite (countable)

Is there a continuous surjective function $f:[0,1] \to [0,1]^2$ such that every level set $f^{-1}(y)$ is a finite set? If the answer is no, what about if we replace the finiteness of level sets by "...
23 votes
3 answers
4k views

Continuous functions taking uncountably many values countably often

Let $f$ be a continuous function defined on the closed interval $[0,1]$. Clearly $f$ is bounded and attains its bounds. Then my question is how often can $f$ take a value in its range countably many ...
-2 votes
2 answers
589 views

Must a countable Polish space be discrete? [closed]

I am looking for an elegant proof of the fact that a countable metric space is complete iff its underlying topology is discrete. It is easy to see that a discrete space is complete because its ...
8 votes
1 answer
362 views

Counting copies of a BA within a BA: arbitrarily many vs infinitely many

Informally, I am wondering if a Boolean algebra $\mathcal{B}$ contains infinitely many disjoint copies of a Boolean algebra $\mathcal{A}$ whenever it contains arbitrarily many disjoint copies of $\...
3 votes
1 answer
141 views

Two paths to the boundary with no holes in between

Let $X\subset \mathbb{R}^2$ be open connected (and let's say bounded), let $x\in X$ and $y\in\partial X$ be such that there is a Jordan curve $\gamma:[0,1]\to X\cup\{y\}$ such that $\gamma(0)=x$ and $\...
10 votes
2 answers
348 views

Source on smooth equivalence relations under continuous reducibility?

This question was asked and bountied at MSE, but received no answer. In the context of Borel reducibility, smooth equivalence relations (see the introduction of this paper) are rather boring since ...
0 votes
1 answer
101 views

If a topological space (X,T) is completely normal, and if we double the point of X, is the resulting space also completely normal? [closed]

I have a question on my proof of the following lemma, and I'd like to know if my answer is correct. Lemma. Suppose $(X,T)$ is any completely normal topological space. Let's double the points of $X$, ...
2 votes
0 answers
242 views

How to compute fundamental groups of slice disk complements?

To compute the fundamental group of the complement $S^3 \setminus K$ of a knot, one usually uses the Wirtinger algorithm. Is there a similarly well-established procedure for computing the fundamental ...
4 votes
1 answer
170 views

A map into a Hilbert space with prescribed orthogonality

Let $X$ be a locally compact separable metric space, and let $L:X\times X\to \mathbb{C}$ be continuous and such that $L(x,x)=1$ and $L(y,x)=\overline{L(x,y)}$, for every $x,y$. Does there always ...
20 votes
3 answers
2k views

Stone–Čech Compactification of $\mathbb{Z}$ with Fürstenberg Topology

The Stone–Čech Compactification of $\mathbb{N}$ as a discrete space has been extensively studied and can be represented using ultrafilters. Consider $X=(\mathbb{Z},\mathcal{T})$, where $\mathcal{T}$ ...
2 votes
1 answer
138 views

Explicit construction of a convex metric

Let $(X,d)$ be a compact, connected, locally connected, locally compact metric space. A result of Bing and Moise (independently) states that $(X,d)$ admits a topology preserving convex metric i.e., ...
3 votes
1 answer
165 views

Reversal of open cover with topologically transitive dynamical system

Let $X$ be a separable metric space, $\phi\in C(X,X)$ be a topologically transitive dynamical system, and $V$ be a non-empty open subset of $X$, and $\nu$ be a locally-positive and atomless Borel ...
9 votes
1 answer
1k views

Is the Milnor construction contractible

Let $G$ be any topological group. Then we can form the infinite join $E_G$ of $G$, i.e. the colimit $G*G*G\cdots$. Is $E_G$ contractible? I mean it is clear that $E_G$ is weakly contractible, but ...
0 votes
1 answer
89 views

Topologically transitive dynamical system mapping space into ball

Let $X$ be a separable Hausdorff topological space and $\phi \in C(X,X)$ be a topologically transitive map. Further, let $V$ be a fixed non-empty open subset of $X$. Then does there necessarily ...
7 votes
1 answer
537 views

What is the topology on the set of field orders

Inspired by this question I was wondering whether there is a natural topology on the set of all orders on a field (that extend a given order on a subfield)? For example for the function field $\...
9 votes
1 answer
4k views

What are some characterizations of the strong and total variation convergence topologies on measures?

I asked this question on StackExchange a few days ago but didn't get any response, so I thought I would try here. The Wikipedia article on convergence of measures defines three kinds of convergence: ...
2 votes
1 answer
470 views

Topology: what defines (non-trivial) paths as being the same trace (curve)?

I posted this originally at MathSE but haven't had any feedback and hope for better luck here. I think I know the answer but can't prove it. Assume that $Y$ is a Hausdorff space and firstly that $p,...
17 votes
6 answers
2k views

The reals as continuous image of the irrationals

In the Wikipedia article about descriptive set theory I read that $\mathbb{R}$ (with its usual topology) is a Polish space, and that every Polish space 1) can be obtained as a continuous image of ...
5 votes
3 answers
607 views

characterization of the unit disk

We know of a charcterization of spaces homeomorphic to [0,1], as being metric continua with 2 noncut points. We have as well a characterization of spaces homeomorphic to the unit circle. I can't find ...
0 votes
1 answer
130 views

A closed subset $B$ of the Hilbert cube such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set

$\def\cube{\mathbf Q}$Let $\cube$ be the Hilbert cube. Give an example of a closed subset $B$ of $\cube$ such that $\operatorname{Int}(B) = \emptyset$ and $B$ is not a z-set. If anyone has any idea ...
10 votes
1 answer
613 views

Contractible and Delta-generated implies strong deformation retract to a point?

If a CW-complex is contractible, then it strongly deformation retracts onto the inclusion of a point. However for general spaces it is well-known that just because a space is contractible, it does ...
5 votes
1 answer
643 views

Restrictions of a local diffeomorphism

I am wondering if a local diifeomorphism has the following property (prove or disprove): Let $M,N$ be differentiable manifolds, and $f:M \to N$ be a local diffeomorphism. Suppose $Z$ is a closed ...
5 votes
1 answer
213 views

One-dimensional compacta as projective limits

Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1. Is it possible to express $X$ as a filtering projective limit of finite graphs? Here finite graphs ...
1 vote
0 answers
116 views

Stone duality- a modification

Let $2$ be the discrete topological space with two elements. For a map of sets $$\beta : X \times Y \rightarrow 2 $$ We get a topology on $X$ and a topology on $Y$. The topology on $X$ is the weakest ...
6 votes
2 answers
397 views

Do mixing homeomorphisms on continua have positive entropy?

I am trying to understand relations between various measures of topological complexity. I have read that expansive homeomorphisms on continua, for example, have positive entropy. But I do not know ...
1 vote
1 answer
994 views

A new generalisation of dimension? part 2

I worked this theory : A new generalization of the dimension? I have a theorem about dimensions which is more general and simple than for matroids. Definition 1: A structure $S$, is a pair $(X, \...
0 votes
1 answer
144 views

About the finished, $\aleph_0$...-compactness

Definitions : $(E,d)$ a metric space is finished-compact if any covering of $E$ by open, we can extract a finite subcover $(E,d)$ is $\aleph_0$-compact if for any infinite covering of $E$ by open, we ...
36 votes
1 answer
3k views

Is there a general theory of "compactification"?

In various branches of mathematics one finds diverse notions of compactification, used for diverse purposes. Certainly one does not expect all instances of "compactification" to be specializations of ...
26 votes
3 answers
2k views

When does a Galois connection induce a topology?

Let $(X,\leq)$ and $(Y,\leq)$ by partially ordered sets. Recall that a(n antitone) Galois connection between $X$ and $Y$ is a pair of order-reversing maps $\Phi: X \rightarrow Y, \ \Psi: Y \...
2 votes
1 answer
152 views

How to define "interior" for the unit arc? [closed]

Let the unit arc be, $$\{x \in \mathbb{R}^2| x_1^2 + x_2^2 =1, x_1 \geq 0, x_2 \geq 0\}$$ There is something I found curious about the unit arc which is that, It has an empty interior viewed as a ...
6 votes
0 answers
191 views

Spaces where the Banach-Mazur game is undetermined

Let $X$ be a non-empty topological space. The Banach-Mazur game on $X$, $\textsf{BM}(X)$, is played as follows: Players I and II play an inning per positive integer. In the $n$-th inning Player I ...
3 votes
3 answers
299 views

Continuous map on the simplex (applicable to fair division)

Let $g$ be a continuous function from the unit simplex $D(n)$ (with $n$ vertices) into itself, that leaves invariant its vertices, and such that $g$ is not onto: to fix ideas say that $g(D(n))$ does ...
15 votes
0 answers
714 views

Is this "Homology" useful to study?

In the usual singular homology of a topological space $X$, one consider the free abelian group generated by all continuous maps from the standard simplex $\Delta^{n}$ to $X$. Now we can ...
10 votes
2 answers
431 views

Which points in the Samuel compactification of a metric space $X$ are limits of uniformly discrete subsets of $X$?

Given a metric space $(X.d)$ the Samuel compactification of $X$, written $sX$, is the unique compactification with the property that if $Y$ is an arbitrary compact Hausdorff space and $f:X\rightarrow ...
-2 votes
1 answer
128 views

$G$- space is locally compact [closed]

Suppose $X$ is a topological space ,$G$ Is a locally compact group.If the quotient space $G\backslash X$ is compact,can we deduce that $X$ is locally compact?
0 votes
0 answers
59 views

Specific property of borelian sigma-algebras

Let X be a set and S a sigma-algebra on X. Let us name borelian sigma-algebra on X a sigma-algebra that is generated by a topology T on X. Given that it is possible for a set X that some sigma-...
1 vote
1 answer
191 views

$\mathbb{R}$-like spaces [closed]

Let us call a topological space $(X,\tau)$ $\mathbb{R}$-like if it is homogeneous, connected, $T_2$, has a basis consisting of open sets homeomorphic to $X$, and $|X|>1$. What is an example of an $...
32 votes
4 answers
5k views

Does the Brouwer fixed point theorem admit a constructive proof?

Wikipedia and a few websites (and a few mathoverflow answers) say there is a constructive proof of the Brouwer fixed point theorem, some others say no. The argument for a constructive proof is always ...
1 vote
0 answers
30 views

The cellularity of the composition of cellular maps (with arbitrary CW decompositioning)

Is the composition of cellular maps cellular? Related to this, I have another question. (I apologize to asking very similar question.) Let ${\sf CWcpx}$ be the category of CW complexes and let ${\sf ...

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