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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.

Everything involving general topological spaces: generation and description of topologies; open and closed sets, neighborhoods; interior, closure; connectedness; compactness; precompactness; separation axioms;countability axioms; bases; convergence: sequences, nets and filters; continuous functions; compactifications; function spaces; proximity; etc. Please use the more specific tags, , , , , whenever appropriate.

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Closure in box and product topology

Let $\mathbb R^\infty$ be the subset of $\mathbb R^\omega$ consisting of all sequences that are eventually zero that is all sequences $(x_1,x_2,\ldots)$ such that $x_i\neq 0$ for only finitely many values of $i$. What is the closure of $\mathbb…
Rutherford Mark
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Show that the boundary of a set equals the boundary of its complement

$\newcommand{\bdy}{\operatorname{bdy}}$ I'm trying to show that $\bdy(A) = \bdy(A^c)$. I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know…
therexists
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Pullback of a covering map

Is the pullback of a covering map $p:E\rightarrow B$ along a continuous map $f:C\rightarrow B$ a covering map?
J.J.
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Difference of closures and closure of difference

Let $A,B$ be subsets of a topological space $X$. Is it true that $\overline{A}-\overline{B}\subseteq\overline{A-B}$? Suppose $x\in\overline{A}-\overline{B}$. So all open sets containing $X$ also contains an element of $A$. And there exists an open…
Paul S.
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Is a function $\mathbb{R}^n \to \mathbb{R}$ which has a closed and connected graph necessarily continuous?

It has been proved here and here that a function $\mathbb{R} \to \mathbb{R}$ which has a closed and connected graph is continuous. This fact is also proved in a nice article by Burgess. I don't know how to generalize these proofs to the case of a…
Geoffrey Sangston
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Does the limit of a descending sequence of connected sets still connected?

Given a descending sequence of sets $$ F_1\supset F_2\supset\cdots F_n\supset\cdots $$ in which each $F_i$ is connected. I wonder if the limit set $$ F=\bigcap_{i=1}^\infty F_i $$ is still connected? I believe it is, but cannot make a proof.…
hxhxhx88
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Compactness of a Sphere

This fact is mentioned liberally in literature along with subsequent mention of Heine-Borel and I am trying to get my head around it. What would be a formal proof of this if we take for example, a unit sphere in $\mathbb R^3 $? In addition, how…
HeinousBarrel
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What is the reason for the term "zero dimensional" in the context of topology?

A topological space is zero dimensional if, and only if, it has a basis consisting of sets which are both open and closed (that is, "clopen"). This definition is according to "Counterexamples in Topology" by Steen and Seebach, 2nd ed. 1978. I see…
Prime Mover
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Showing that every connected open set in a locally path connected space is path connected

I'm trying to figure out whether my proof is correct for a question I'm trying to tackle in Topology by James R. Munkres. Task: Let $X$ be locally path connected. Show that every connected open set in $X$ is path connected. My attempt at a proof:…
Libertron
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Topological characterization of the closed interval $[0, 1]$.

I would like to learn purely topological characterizations of the closed real intervals (to justify the existence of algebraic topology). In particular, such a characterization should not use real numbers. I would like to see in what sense $[0, 1]$…
Alexey
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How to prove that $ \text{int}(\text{cl}(A)) = \text{cl}(\text{int}(A)) $?

How can one prove that $ \text{int}(\text{cl}(A)) = \text{cl}(\text{int}(A)) $, where $ A \subseteq \mathbb{R}^{n} $? This is true for all sets that I have tried, but I can’t prove it formally.
Ashot
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A five-part problem that uses ends and the Cantor set to prove that there are $c$ non-homeomorphic connected open subsets of $\mathbb{R}^2$

My question comes from Spivak's "Comprehensive Introduction to Differential Geometry Vol 1" (It's Chapter 1, Problem 24). Background: Let $X$ be a connected, locally connected, locally compact, and hemicompact Hausdorff space. And end of $X$ is…
alephzero314
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Proof that an infinite product of discrete spaces may not be discrete

I am trying to prove that an infinite product of discrete spaces may not be discrete. I tried taking the simplest nontrivial discrete space, $X:=\{0,1\}$ with the discrete topology, and tried to find a sequence of $0$s and $1$s in…
Alex Petzke
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Quotient Space $\mathbb{R} / \mathbb{Q}$

I've just learned about topological quotient spaces and was wondering if anyone can help me with this example I thought of. Let $(\mathbb{Q}, +)$ be the usual group of rational numbers for addition, likewise $(\mathbb{R}, +)$. Set $S$ to be the set…
jvr
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Sequentially continuous implies continuous

In topological spaces, which condition is necessary for a sequentially continous function $f: (X,\tau_x) \rightarrow (Y,\tau_y)$ to be continous? I have tried to prove this making the space X be $T_1$ and then making it Hausdorff but I don't get the…
Mihael Arce Baldo
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