Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [gn.general-topology]

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

4
votes
1answer
144 views

When Stone–Čech compactification is totally disconnected

A topological space $X$ is totally disconnected if the connected components in $X$ are the one-point sets, and a topological space, $X$ is called completely regular exactly in case points can be ...
3
votes
1answer
494 views

extending continuous functions from dense subsets to quasicompacts

I am interested under what assumptions one can always extend continuously a function defined on a dense subset; the range of the function is compact but not necessarily Hausdorff. That is, I am ...
1
vote
0answers
40 views

Difference between planar sub-continua and sub-continua on the surface $\mathbb{T}^2$?

Can anyone tell me what is the essential difference between planar sub-continua and sub-continua of the torus? I will appreciate if you can give me some references.
3
votes
2answers
856 views

Finite Topology vs sigma Field

Suppose we have a finite $\sigma$ -field $S$, of which $A$ and $B$ are member sets. Since $S$ is closed under union and complementation [by definition], it follows that $(A' \cup B')' = (A \cap B)' \...
3
votes
2answers
137 views

Is a plane set still metrizable if two new subsets are declared open?

I am thinking of forming a finer topology on a particular subset of the plane. Let $X\subseteq \mathbb R ^2$ be endowed with the Euclidean topology $\tau$. Let $A,B\subseteq X$. Let $\tau'$ be the ...
9
votes
1answer
187 views

Rothberger property for finite covers

Let us recall that a topological space $X$ has the Rothberger property if for any sequence $(\mathcal U_n)_{n\in\omega}$ of open covers of $X$ there exists a sequence $(U_n)_{n\in\omega}\in\prod_{n\...
6
votes
1answer
133 views

Continuous binary operations on $\beta\mathbb{N}$

It is well-known that the operation of addition of two ultrafilters on the set $\mathbb{N}$ of natural numbers which extends the natural addition on $\mathbb{N}$ to $\beta\mathbb{N}$, the Cech-Stone ...
1
vote
1answer
88 views

connected and quasi-connected separators of a space

Does there exist a connected topological space $X$ and a subset $A\subseteq X$ such that no connected component of $A$ separates $X$, but some quasi-component of $A$ separates $X$? Meaning $X\...
10
votes
0answers
258 views

If an additive group of $\Bbb R^2$ contains a smoothly deformed circle, is it necessarily all of $\Bbb R^2$?

It can be shown that if an additive subgroup of $\Bbb R^2$ contains the unit circle, then it is necessarily all of $\Bbb R^2$. Does this also hold for a suitably smoothly deformed unit circle? ...
1
vote
2answers
99 views

Slightly finer topology vs a quasi-component

Let $(X,\tau)$ be a topological space, and let $Q$ be a quasi-component of $X$. Let $S$ be a subset of $X\setminus Q$. Then is $Q$ necessarily a quasi-component of $X$ in the topology generated by $\...
3
votes
1answer
175 views

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 ...
2
votes
1answer
222 views

Surjectivity of self-isometries as property of metric spaces

A metric space $(X,d)$ is said to be bounded if there is $r\in\mathbb{R}$ such that for all $x,y\in X$ we have $d(x,y) \leq r$. A self-isometry is a map $\iota:X\to X$ such that for all $x,y\in X$ we ...
3
votes
0answers
69 views

Name for mappings that are “not quite projections”

Is there a known name for the following definition? Consider topological spaces $X$, $Y$ and $f: X \rightarrow Y$ a continuous mapping. Then, $f$ is an "almost projection" if there is a topological ...
3
votes
0answers
99 views

Embeddability into $\beta\omega$ and $\omega^*$

It is well known that under CH every totally-disconnected compact F-space of weight at most $\omega_1$ can be embedded into the remainder $\omega^*=\beta\omega\setminus\omega$ of the Cech-Stone ...
2
votes
1answer
103 views

Non-discrete $T_2$-space $(X,\tau)$ with $2^{|X|}$ retracts

If $(X,\tau)$ is a topological space, we call $A\subseteq X$ a retract if there is a continous map $r:X\to A$ such that $r(a) = a$ for all $a\in A$ (we assume $A$ to be endowed with the subspace ...
2
votes
1answer
68 views

$T_1$ version of Engelking theorem?

Theorem 6.1.23 in Engelking's Topology book says that in a compact space $X$ each quasi-component is connected. Quasi-component means the intersection of all closed-and-open subsets of $X$ containing ...
3
votes
1answer
132 views

Extending continuous functioms defined on the irrationals

Lavrentieff proved a Theorem which implies that every real valued continuous function defined on a dense subset $D\subseteq \mathbb R$ admits a continuous extension to some $G_\delta $ subset of $\...
3
votes
0answers
91 views

Slightly finer topology on a connected space

Let $(X,\tau)$ be a connected Hausdorff space. Suppose $S\subseteq X$ is such that for every $U\in\tau$, $$U\cap S\neq\varnothing \implies U\cap \overline S\setminus S\neq\varnothing.$$ Is it ...
1
vote
1answer
135 views

Descending almost-contained subsets of $\omega$ [closed]

Let $A$ be an infinite subset of $\omega$ such that $\omega\setminus A$ is also infinite. Under the Continuum Hypothesis is there a sequence $(A_\alpha)_{\alpha<\omega_1}$ of subsets of $\omega$ ...
2
votes
1answer
143 views

What is the topological/smooth analogue of Nagata compactification

A celebrated theorem of Nagata and subsequent refinements to schemes and algebraic spaces say that over a not-completely-monstrous base scheme, any separated morphism can be openly immersed in a ...
3
votes
0answers
75 views

Topological characterization of closed surfaces

I am looking some reference, if it exist, that generalized the Moore`s characterization of the 2-sphere. To be more precise, Moore characterized 2-sphere by these two axioms: A space X is a 2-sphere ...
3
votes
0answers
136 views

Compactification of Tychonoff spaces without full axiom of choice

If $X$ is a Tychonoff space, then using the Tychonoff theorem and thus the full axiom of choice, it follows that $X$ admits a Hausdorff compactification. My question is : what remains true if we do ...
15
votes
1answer
936 views

Duality between topology and bornology

I want to understand in what sense topology is dual to bornology at a most basic level. Therefore, I rephrased the definition of a bornology in the following way: Let $X$ be a set and let $\mathcal{P}...
50
votes
0answers
1k 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 ...
2
votes
1answer
177 views

Stone-Cech Compactification of the real line

I have a question in $\beta\mathbb{R}$, the Stone-Cech compactification of the real line $\mathbb{R}$. My question is: is $\beta\mathbb{R}$ a $\mathrm{F}$-space, i.e., the closure of two disjoint open ...
11
votes
2answers
354 views

What are the 'wonderful consequences' following from the existence of a minimal dense subspace?

In Peddechio & Tholens Categorical Foundations they quote PT Johnstone in their chapter on Frames & Locales: ...the single most important fact which distinguishes locales from spaces: the ...
32
votes
11answers
3k views

Continuous relations?

What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful? I ...
4
votes
0answers
104 views

Maximal connected subtopologies

This is related to an older question. Let $(X,\tau)$ be a topological space. Trivially, the indiscrete topology $\{\emptyset, X\}$ is a connected subtopology of $\tau$. Is there a connected topology ...
5
votes
1answer
333 views

Base zero-dimensional spaces

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\mathcal ...
0
votes
0answers
55 views

Complement of a meagre set in a Baire space

I remember reading somewhere that the complement of a meagre set in a Baire space is also a Baire space and this is in fact easy to prove. Looking for this result in the standard collection of ...
-4
votes
1answer
139 views

Topological spaces without retracts [closed]

Is there a way to see whether a topological space $\Omega$ does not allow retractions $r: \Omega \mapsto B$, with $B$ a given subspace of $\Omega$ ? In other words: when is a space not retractable ...
29
votes
1answer
776 views

If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?

I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
3
votes
1answer
103 views

Separable topology on a group

In the paper "Continuous isomorphisms onto separable groups", Applied General Topology, (13) 2012, 135--150, L. Morales Lopez proved Theorem: Let $G$ be an Abelian group with $|G| \leq 2^{2^{\...
2
votes
1answer
76 views

Under what conditions the End-compactification is metrizable

Suppose that $X$ is a hemicompact space, connected and locally connected. In that case, it seems that it is possible to define a "End-compactification" of $X$ (in the sense of Freudenthal). Suppose ...
21
votes
2answers
3k views

complement of a totally disconnected closed set in the plane

While preparing a course in complex analysis, I stumbled over a remark in Dudziak's book on removable sets, namely that any totally disconnected $K \subset\subset {\mathbb C}$ must have a connected ...
11
votes
1answer
316 views

Do solenoids embed into Möbius strips?

I found a strange attractor which looks a lot like a solenoid. The attractor continuum is the closure of a continuous line which limits onto itself, and it is locally homeomorphic to Cantor set times ...
12
votes
1answer
255 views

“Scott completion” of dcpo

If $A$ is poset with all directed suprema, it is common to consider the Scott topology on $A$, whose open subsets are the $U \subset A$ such that $U$ is upward closed and if $\bigcup_I a_i \in U $ for ...
31
votes
6answers
11k views
5
votes
1answer
207 views

On filters possessing a countable network

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$ A family $\mathcal N$ of subsets of $\omega$ is called a network ...
1
vote
0answers
54 views

Descent data and trivialization of bundles via coherent isomorphisms of fibers

In this MO question I tried to understand how a trivialization of a bundle (continuous map) $\begin{smallmatrix}A\\ \downarrow\\ B \end{smallmatrix}$ is related to a coherent family of isomorphisms ...
1
vote
0answers
96 views

Mackey topology characterising property

Let $V$ be a topological $k$-vector space. Let $V^{\star}$ denote the space of all linear functionals $V \rightarrow k$ and $V' \subset V^{\star}$ the subspace of all continuous linear functionals. ...
6
votes
0answers
99 views

The automorphism group of the fibered cylinder

My collegue (Oleg Gutik) is interested in finding a proper reference to a description of the group $G$ of homeomorphisms $h:\mathbb T\times\mathbb R\to\mathbb T\times\mathbb R$ of the cylinder that ...
2
votes
0answers
45 views

Does there exist a continuous path between two sets of oriented basis for a vector space out of a collection of subspaces?

This is cross post to the question at MSE. Let $V_1, V_2, \dots, V_n$ be a collection of vector subspaces in $\mathbb R^n$. For each $j=1, \dots, n$, $\dim(V_j) = m$ with $2 \le m < n$. We also ...
5
votes
1answer
73 views

continuity of certain map which is defined on a Stonean space

Let $G$ be a discrete group which acts continuously on a Stonean space $\Omega$. Consider the map $f\colon \Omega\to \{0,1\}^G$ sending $x\in \Omega$ to $\chi_{G_x}$, where $\chi_{G_x}$ denotes the ...
6
votes
1answer
181 views

Is each Parovichenko compact space homeomorphic to the remainder of a soft compactification of $\mathbb N$?

Definition 1. A compactification $c\mathbb N$ of the discrete space $\mathbb N$ is called soft if for any disjoint sets $A,B\subset\mathbb N$ with $\bar A\cap\bar B\ne\emptyset$ there exists a ...
3
votes
3answers
174 views

If $X$ has the fixed point property, what about $\text{Cont}(X,X)$?

If $(X,\tau)$ is a topological space, we denote by $\text{Cont}(X,X)$ the collection of all continous functions $f:X\to X$. We say that $(X,\tau)$ has the fixed point property if for any $f\in\text{...
1
vote
0answers
75 views

Cobordism of an annulus with a non-vanishing vector field

Let $M$ be a compact three-dimensional manifold with corners, which is a cobordism of the two-dimensional annulus. In particular, the codimension one boundary of $M$ consists of two copies of the ...
4
votes
1answer
171 views

Does the notion of a compactly generated space (or $k$-space) depend on the choice of universe?

We recall the notion of a $k$-space (or compactly generated space) to fix our notations. For every topological space $X$, we can define a category $\mathfrak{M}_X$. The class of objects of $\mathfrak{...
11
votes
1answer
222 views

The Parovichenko cardinal, is it equal to $\max\{\aleph_2,\mathfrak p\}$?

Let us define the Parovichenko cardinal $\mathfrak{P}$ as the largest cardinal $\kappa$ such that each compact Hausdorff space $K$ of weight $w(K)<\kappa$ is the continuous image of the remainder $...
9
votes
1answer
309 views

Noetherian spectral space comes from noetherian ring?

Let $X$ be a spectral space (en.wikipedia.org/wiki/Spectral_space), i.e. a space of the form $\textrm{Spec}(A)$ for some commutative ring $A$. If $X$ is noetherian, does there also exist a noetherian ...