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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.

4
votes
1answer
149 views

Rays in non-compact spaces

According to the book Infinite homotopy theory by Baues-Quintero, if $X$ is a locally compact, locally connected, connected metrizable space, its Freudenthal ends can be identified (Proposition 9.20) ...
5
votes
1answer
134 views

Are there any non-trivial convergent sequences in the maximal ideal space of the measure algebra?

Consider the measures on the circle, $M(\mathbb T)$, endowed with the convolution product which makes it a unital Banach algebra under the total variation norm. Denote by $\Delta$ the maximal ideal ...
4
votes
2answers
189 views

Doubling dimension vs other metric dimensions

For separable metric spaces, three fundamental notions of dimension are equivalent: $$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$ Where does the doubling dimension fit into the picture?
3
votes
1answer
123 views

Trees and Shabat polynomials

Recently, I read the relation between Shabat polynomials and trees. The book [0] says that if $p: \mathbb{C} \to \mathbb{C}$ is a degree-$n$ polynomial such that the segment $[-1,1]$ contains no ...
2
votes
3answers
209 views

Example of an $\omega_1$ decreasing chain of dense semicontinua?

In his well-known paper Bellamy constructs an indecomposable continua with exactly two composants. The setup is as follows: We have an inverse-system $\{X(\alpha); f^\alpha_\beta: \beta,\alpha < \...
3
votes
1answer
270 views

Is $\Box_{n\in\omega}[0,1]$ connected?

Let $\Box_{i\in I} X_i$ denote the box product of the spaces $X_i$. The box product $\Box_{n\in\omega}\mathbb{R}$ is not connected, since the collection of bounded sequences is both open and closed. ...
3
votes
1answer
111 views

Existence of a discrete subset

Let $X$ be a topological space. $Y$ is a discrete subset of $X$ if it has a discrete topology induced by the topology of $X$. This is equivalent to the fact that for every $y\in Y$ there is an open $U\...
5
votes
1answer
189 views

How many disjoint compact sets are needed to form a connected compactum?

Let's assume all spaces are metrizable. For each connected compact space $X$ let $\mathscr K(X)$ be the set of all partitions of $X$ into non-empty compact sets, excluding the trivial partition $\{X\}...
3
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0answers
103 views

Topology Data Analysis - faster algorithm

The Topology Data Analysis uses the Mapper algorithm, but computational complexity is not good. Is there an alternative algorithm for algorithm Mapper? Is there an algorithm that works faster?
3
votes
1answer
150 views

Fixed-point property and $T_0$ separation property

Each topological space $A$ with fixed-point property is $T_0$ space. Proof: suppose, two different points $a_1$ and $a_2$ belong to the same open subsets of $A$. Then the function $$f(a)= \begin{cases}...
3
votes
0answers
119 views

One strong fixed-point property

Each topological space $A$ with fixed-point property is connected (all clopen subsets are trivial). This is an analog of Rice theorem (all decidable subsets are trivial). Suppose, we have a space $A$ ...
2
votes
1answer
73 views

Every $b$-discrete space $X$ with countable injective weight is basically disconnected?

Recall that a space $X$ is called basically disconnected [1] if every cozero-set has an open closure. According to Tkačuk [2], a space $X$ said to be $b$-discrete if every countable subset of $X$ is ...
2
votes
0answers
55 views

Enveloping a Jordan curve with a trace of another one

This question is inspired by this one, or rather the way I understood it. Let $\gamma$ and $\delta$ be parametrised Jordan curves on the plane (i.e. homeomorphisms from $S^1$ onto its image in $\...
4
votes
1answer
158 views

Is $\mathbb{Q}$ the continuous image of a Golomb-like space, or vice versa?

The set $${\cal B} = \big\{\emptyset\big\}\cup\big\{\{a + bn: n\in\omega\}: a\in\omega, b\in(\omega\setminus\{0\})\big\}$$ is a basis for a topology $\tau$ on $\omega$. Is there a surjective ...
4
votes
1answer
108 views

Approaching a space with a ray

Given a separable metric space $X$, what are some ways of forming a new metric space $Y$ such that: (i) $Y$ contains a ray $R\simeq [0,\infty)$ ($\simeq$ means homeomorphic); (ii) $R$ is open and ...
5
votes
1answer
280 views

Minimal ideals of the ring of continuous functions

A minimal ideal of a commutative ring $R$ is a nonzero  ideal which contains no other nonzero ideal.  Let $X $ be a completely regular topological space and $C (X) $ the ring of all real valued ...
9
votes
1answer
251 views

What is the (genuine) name for the Gutik hedgehog?

Working with non-regular topological semigroups, my collegue Oleg Gutik discovered a special space $H$ which we named Gutik's hedgehog. It is homeomorphic to the space $$H:=\{(0,0)\}\cup\{(\tfrac1n,0):...
1
vote
1answer
91 views

Non-uniqueness in Krylov-Bogoliubov theorem

So apparently the Krylov-Bogoliubov theorem says that every continuous function $f:X\to X$ on a compact metrizable space $X$ has an invariant probability measure $\mu$. Of course, if $X$ is just a ...
2
votes
2answers
316 views

“Minkowski Multiplication” of Convex Sets?

Apologies if this question might be trivial or has been asked already (haven't found an equivalent post), but I am trying to figure out whether the following is true: Given two convex sets $\mathcal{...
1
vote
0answers
61 views

Showing that $b$ is a inner point of $\mathcal{G}$ where $\mathcal{G}$ is a subset of $\mathbb{R}^{N+3}$ determined by $\mathcal{M}^{+}$

Let $(\Xi,\mathscr{E})$ be a measurable space, $(\mathbb{R_{+}},\mathfrak{B})$ other measurable space where $\mathfrak{B}$ a $\sigma$-algebra. We consider the measurable space $(\Xi\times\Xi\times\...
7
votes
2answers
536 views

Reflection of the chain-condition: clarifying a suspected typo

In the 2002 paper Reflecting Lindelöfness in Topology and its Applications, J. Baumgartner and F. Tall state that "by an easy Löwenheim–Skolem argument," an uncountable (say $T_2$ or $T_3$) ccc ...
0
votes
0answers
124 views

a property for real valued functions

Let $f$ be a real valued continuous function over a completely regular topological space $X$, I am looking for a condition on $f$ such that if there exist real valued continuous functions $g_1,...,...
13
votes
1answer
394 views

Continuum Hypothesis and the fact that every co-finite topological space, with uncountable underlying set , is contractible

Let $X$ be a co-finite topological space. If $|X| \ge 2^{\aleph_0}=\mathfrak c$, then $X$ is contractible (https://en.wikipedia.org/wiki/Contractible_space) . Indeed, there is a bijection $f: X \times ...
14
votes
2answers
383 views

Proper topological spaces

Recall that a topological space is ccc, or has the countable chain condition, if every family of pairwise disjoint open sets is countable. But equivalently, we can say that the forcing defined with ...
3
votes
1answer
125 views

Nonmetrizable Corson compacta with ccc

It is known that under $MA+ \neg CH$, every Corson compact space with the countable chain condition (ccc) is merizable. It is also known that, under $CH$, there exist nonmetrizable Corson compact ...
6
votes
1answer
426 views

What is “topology in dimension 3.5”?

I've noticed a couple of conference titles which reference something called "topology in dimension 3.5," such as this one and this one. This subject seems quite mysterious to me — it looks like ...
2
votes
1answer
60 views

Haar-$\mathcal{I}$ set and Polish groups

Let $\mathcal{I}$ be a semi-ideal of sets with empty interior on a compact metrizable space $K$. Let an $F_σ$-set $A$ in a Polish group $X$ generically Haar-$\mathcal{I}$. Then is $A$ always ...
5
votes
0answers
147 views

Local isometry of complete length spaces that is not a covering map

Let $\pi:\widetilde{M}\to M$ be a surjective local isometry between complete length spaces (local isometry means that every point $x\in \widetilde{M}$ has a neighborhood which is isometrically mapped ...
8
votes
0answers
107 views

“Generic properties” of open neighborhood boundaries in compact metric spaces

Suppose we have a compact metric space $X$ with some designated point $a$ and closed set $B$ such that $a\notin B$. Let $A_0=\{a\}$ and $B_0 = B$. We'll play a game where on each player's turn they ...
13
votes
1answer
287 views

Continuous images of $\beta \mathbb{N} \setminus\mathbb{N}$

Let $\beta \mathbb{N}$ denote the Stone-Cech compatification of the natural numbers and $\beta \mathbb{N} \setminus\mathbb{N}$ denote the reminder of this compactification. I wonder if there is a ...
2
votes
2answers
190 views

Obstructions to being a trivial line bundle over itself

Let $X$ be a geodesic metric space. Are there known local obstructions to the existence of a (bi-Lipschitz) homeomorphism between $X$ and $X\times \mathbb{R}$?
3
votes
0answers
132 views

Tietze extension theorem for lower semi continuous functions

On the Tietze extension theorem, if instead of a continuous function "f" we use a lower semi continuous function on a closed subspace of a metric space, is the theorem correct? I mean, can we extend ...
0
votes
1answer
134 views

does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and ..?

Suppose $(X,d)$ is a metric space and $f:[0,1] \rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, does there always exists a path $g:[0,1] \rightarrow X$ from $f(0)$ to $f(1)$ that ...
2
votes
0answers
45 views

When $C_k(X,2)$ is Baire?

$C_k(X,2)$ is the family of all continuous function from $X$ to $\{0,1\}$ with compact-open topology. for which kind of topological spaces $C_k(X,2)$ is Baire or meager? Or is there something ...
4
votes
2answers
221 views

Map between manifolds and open dense subsets

Let $X$,$Y$ be compact, connected, smooth manifolds of the same dimension. Suppose you have a surjective smooth map $f : X \rightarrow Y$, such that $|f^{-1}(p) | \leq k$ for all $p \in Y$. Let $U \...
7
votes
0answers
94 views

The first homotopic Baire class

Let $X$ and $Y$ be topological spaces. A map $f:X\to Y$ belongs to the first Baire class (to the first homotopic Baire class), if there exists a continuous map $H:X\times \omega\to Y$ (a continuous ...
9
votes
3answers
364 views

Does the lattice of all topologies embed into the lattice of $T_1$-topologies?

Let $\kappa$ be an infinite cardinal, and let $\text{Top}(\kappa)$ be the lattice of all topologies on $\kappa$, ordered by $\subseteq$. Let $\text{Top}^{T_1}(\kappa)$ be the lattice of all $T_1$-...
2
votes
0answers
76 views

Union of Two Faces, using the Jordan Curve Theorem

Consider four disjoint points in the plane, $v_{1}$, $v_{2}$, $v_{3}$ and $v_{4}$. The cycle, $C:=v_1v_2v_{3}v_{4}v_{1}$, is the union of the (Jordan) arcs, $A_{12}$, $A_{23}$, $A_{34}$, and $A_{41}$, ...
1
vote
1answer
83 views

Connected $T_2$-space such that not all closed subsets are fibers

If $(X,\tau)$ is a topological space, then we say $A\subseteq X$ is a fiber if there is $f:X\to X$ continuous and $y\in X$ such that $A = f^{-1}(\{y\})$. For any $T_1$-space it is clear that fibers ...
6
votes
1answer
196 views

Is there a compactification with nontrivial connected remainder?

Question: Let $X$ be a continuum and $p \in X$. Under what conditions does there exist a compactification $\gamma (X-p)$ with $\gamma (X-p) - (X-p)$ connected and nondegenerate? Throughout, $X$ is a ...
6
votes
2answers
237 views

Infinite “almost rigid” homogeneous $T_2$-space

A topological space $(X,\tau)$ is said to be homogeneous if for all $x,y$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there an infinite homogeneous Hausdorff space $(X,\...
2
votes
0answers
91 views

Homogeneity of the space of semicontinuous functions

I am interested in the topological homogeneity of function spaces. Question. Let $X$ be a Tychonoff space, let $USC(X)$ be a space of upper semicontinuous functions on $X$ and let $USC(X)^+$ be a ...
5
votes
1answer
161 views

Characterizing topological spaces $X,Y$ whose function space $C_k(X,Y)$ is Baire

I am looking for a characterization of topological spaces $X,Y$ for which the function space $C_k(X,Y)$ is Baire. Here $C_k(X,Y)$ is the space of continuous functions from $X$ to $Y$, endowed with ...
6
votes
2answers
210 views

Does $\aleph_0$-density of regular open algebra entail existence of countable basis?

Suppose that the family $\mathrm{RO}(X)$ of regular open subsets of $(X,\mathscr{O})$ is a basis of $X$. Let the density of $\mathrm{RO}(X)$ (considered as boolean algebra) be $\aleph_0$. Does $X$ ...
13
votes
1answer
335 views

Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects

These are five important constructions and I would like to know how they are related. The $n$th unordered configuration space of a space $X$ is $$ \operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
0
votes
1answer
117 views

A property of compact topological space via certain $C^*$ embedding in operator algebras

Assume that $A$ is a unital $C^*$ algebra. Is there a $C^*$ embedding of $A$ in some $B(H)$ whose image is a hereditary $C^*$ subalgebra of $B(H)$? If not, is the answer affirmative when $A$ is ...
4
votes
0answers
185 views

A kind of 0-1 law?

Suppose that P is a Borel subset of Baire $\times$ Baire, such that for every pair $x,x'$ of reals in the horizontal copy of Baire, if: $x,x'$ are $E_0$-equivalent (that is, $x(n)=y(n)$ for all but ...
3
votes
1answer
112 views

Non-homogenizable topological spaces

A topological space $(X,\tau)$ is said to be homogeneous if for $x,y\in X$ there is a homeomorphism $\varphi: X\to X$ such that $\varphi(x) = y$. Let us call a topological space $(X,\tau)$ ...
2
votes
0answers
75 views

If $H$ and $Z$ are closed subgroups generating $G$, is $H \times Z \rightarrow G$ an open map?

Let $G$ be a Hausdorff topological group with center $Z$ and closed subgroup $H$. Suppose that $H.Z = G$. Is the product map $$H \times Z \rightarrow G$$ necessarily an open map? That is, can we ...
0
votes
0answers
55 views

continuous map from $\mathbb R$ to $\mathbb R^2$ that send any convex on a convex [duplicate]

Let $f$ be a continuous fonction from $\mathbb R$ to $\mathbb R^2$, such that for any $a<b\in \mathbb R,\,\, f([a,b])$ is convex. Is there a line $D\subset \mathbb R^2$ such that $f(\mathbb R)\...