Questions tagged [general-topology]
Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; separation axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; etc. Please use the more specific tags, (algebraic-topology), (differential-topology), (metric-spaces), (functional-analysis) whenever appropriate.
60,025 questions
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Proving the set of zeros of a $F: \mathbb{R}^{n} \to \mathbb{R}^{m}$, $C^{1}$ func is a $n-m$ dimensional manifold with the Implicit function theorem
This is a follow up question to this question I asked earlier today, in which I asked whether the function $g$ we get from the implicit function theorem is a homeomorphism. As pointed out in the ...
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outer regular measure
We say that $(X,\mathcal{M},\mu)$ is topological measure space if $X$ is topological space such that all open sets are in $\mathcal{M}$. We say that $\mu$ is inner regular if$$\mu(A)=\sup\{\mu(K),\...
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When compacts generate the Borel $\sigma$-algebra, is $X$ $\sigma$-compact?
Let $X$ be a Hausdorff topological space and let $\mathcal K$ denote the family of compact subsets of $X$. Assume that the Borel $\sigma$-algebra $\mathcal B(X)$ is generated by compact sets, i.e.
$$
\...
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Topological obstructions to finding a map $g\colon \mathbb{R}^2 \longrightarrow S^2$ subject to constraints
Given a continuous map $f\colon \mathbb{R}^2 \longrightarrow S^2$, is it possible to find a continuous map $g\colon \mathbb{R}^2 \longrightarrow S^2$ such that
$g(x) \neq f(x)$ for all $x \in \mathbb{...
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Finest possible refinement of a topology that does not alter the Krull dimension
While I was thinking about some problem in algebraic geometry I came up with the following construction, and I am asking myself if this has a name.
A (conventional) Google search yielded no results ...
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Well defindness of homotopy map $F(s,t)=h\left(f(s-st)\right)$
Let $I=[0,1]$, $h:(X,x_0) \to (Y,y_0)$ and $[f] \in \pi_1(X_0)$. Can I construct the following map $$F(s,t)=h\left(f(s-st)\right):I \times I \to Y?$$
Here, $F(s,0)=h \circ f(s)$ and $F(s,1)=e_{y_0}$. ...
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Existence of Uncountable Discrete Family induced by Uncountable pw-disjoint Open Sets in a Metric Space
There are lots of questions about the upper separation properties, where people want to get open sets around infinite collections of closed sets. For example:
construction of a discrete family
...
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Conditions for Separable $T_{3\frac{1}{2}}$ Spaces to be Strongly Paracompact
It's known that $T_3$ Lindelof spaces are strongly paracompact, but I was wondering what sorts of conditions are needed to ensure strong paracompactness. For $T_3$ locally Lindelof spaces, ...
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Intuition Regarding Homeomorphism Classes of Quotient Topologies
While I understand formally what quotient topologies and homeomorphisms are, I'm having a hard time intuitively/visually grasping/answering questions like "What common shape in $\mathbb{R}^2$ or $...
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Is the weak sequential topology on a separable Hilbert space always metrizable?
In a separable Hilbert space $X$, consider the weak sequential topology, i.e., the topology whose closed sets are precisely the weakly sequentially closed subsets of $X$.
A central question in ...
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Question on Local Connectedness, Connectedness, and Product Space
The proposition I was trying to prove is as follows:
If $Y = \prod_{n \in \mathbb{N}} X$ and $Y$ is locally connected, then $X$ is connected.
My proof is as follows but I think it has a flaw, but I ...
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Investigating the topological properties of a quotient space of $\mathbb R^2$, given by identification of irrational lines through $0$.
My question is about a very erratic quotient space. I encountered this space in some topology exercise. The space $X$ is described in the following:
Let $\mathbb R^2=\{(x,y):x,y\in \mathbb R\} $ be ...
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Isomorphism between two measure spaces; reference request for Rohlin (1952)
As part of the proof for Theorem 1 of Kamae et al.'s paper, "Stochastic inequalities on partially ordered spaces,"Stochastic inequalities on partially ordered spaces," the following ...
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Structure on a manifold - definition confusion
Last week I started a course discussing Lie groups. So far, we’ve had only one lecture, but it was enough to make my head spin (there are a couple of definitions that I couldn’t get my head around). I ...
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How "close" can we get to a homeomorphism between non-homeomorphic surfaces?
In order to qualify as a homeomorphism, a map between topological spaces must be (1) injective, (2) surjective, (3) continuous, and (4) its inverse must be continuous.
I suspect that these four ...
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Urysohn's lemma for closed $G_{\delta}$ sets
Exercise 33.4 of Munkres asks to prove that given a $G_{\delta}$ closed subset $A$ of normal space $X$, we can find a continuous function $f:X \to \mathbb{R}$ such that $f(x)=0$ for every $x \in A$ ...
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Extension of continuous functions from dense subsets of $\mathbb{R}^T$
Let $T$ be an uncountable index set and consider the product space $\mathbb{R}^T$ with the product topology. Suppose $Y \subset \mathbb{R}^T$ is a dense linear subspace.
Is it true that every ...
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Uniform Compactness Radius for Locally Compact Metric Spaces
I was reading a nice paper, but right at the beginning of one of the main results they claim:
$X$ is a locally compact metric space, so it has an equivalent metric such that every closed ball of ...
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How can $\mathcal{H}$ be a sub-basis if no topology is given?
This is a second follow up question to this question I asked today (this is the first followup question).
In the original question I asked to prove the following:
Let $\left\{ \left(X_{\alpha} , \...
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Generating collection of subsets to the weak topology on a set $Y$ induced by functions from $Y$ to a collection of topological spaces
Definitions
Let $X$ be a set. A collection of subsets $\mathcal{B} \subseteq \mathcal{P} \left( X \right)$ is said to be a basis for a topology on $X$ if:
For every $x\in X$ there exists $B \in \...
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Question About Milnor's proof of the Brouwer's fixed-point theorem
I'm currently going through Milnor's proof of the Brouwer's fixed-point theorem, which I've linked here. I am able to follow the proof up to theorem 2 - afterwards, I've having a bit of trouble ...
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Context on the Arzelà–Ascoli theorem
I am doing an expository presentation on the theorem, and I understand the proof. What I fail to understand is, why is this theorem important? Why does any sequence $\{f_n\}$ containing a uniformly ...
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What properties does the one-point compactification of the Arens-Fort space have?
This is intended to be a self-answering question, which is allowed on StackExchange sites (see here).
We are interested in the traits of the one-point compactification of the Arens-Fort space not yet ...
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Given the space $\mathbb{X} = A\cup B$ with their own topology prove that the piecewise function is $(\tau_x , \tau_y)$ continuous
I'm studying topology and came across this question
Let $(\mathbb{X} , \tau_x) , (\mathbb{Y} , \tau_y)$ topological spaces. $A,B \subseteq \mathbb{X}$ closed subsets of $\mathbb{X}$ such that $\mathbb{...
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Topology and lexicographic order
I am trying to do the following exercise in general topology:
Let $X=\mathbb{R}\times [-1,1]$, with the topology generated by the open intervals with respect to the “lexicographic” order ($(x,y)<(x’...
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