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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.

57719 questions
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If $A\cap B=\emptyset$ and $A$ is an open set, then $\overline A\cap \operatorname{Int} (\overline B)=\emptyset$??

A and B are subsets of a topological space. How do i prove that if $A \cap B = \emptyset$ and A is an open set, then $\overline A \cap Int(\overline B) = \emptyset $? Here's my part of proof: $A \cap B = \emptyset \rightarrow A \cap \overline B = A…
Anton
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If $f: X\to Y$ and $g: Y\to X$ are continuous bijections, not necessarily inverses, is $X\cong Y$

I was thinking about to show two sets are bijective, it suffices to construct an injective function both ways. That is, if $f: X\to Y$ and $g: X\to Y$ are both injective (or both surjective), then there exists a bijection between $X,Y$. But, what if…
user975734
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Line of circles and star made of circles, are they homeomorphic?

How can I determine if this two figures are homeomorphic? I'm guessing they're not homeomorphic. I have tried using cut points but from what I understand both figures have the same number of cut points. I can see that in the first picture the circle…
u2718281
  • 43
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If nonempty, nonsingleton $Y$ is a proper convex subset of a simply ordered set $X$, then $Y$ is ray or interval?

This is from Exercise 7 in p. 92 in Munkres's Topology. Except for the trivial cases such as $Y$ is empty set or singleton, it seems if $Y$ is convex in an simply ordered set $X$ then $Y$ is interval or ray. But I cannot start my proof because I…
le4m
  • 3,006
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Question about almost metrizable space and $k$-spaces.

A space $X$ is said almost metrizable if for every $x\in X$ there exists a compact set $F \subset X$ with a countable neighbourhood base in $X$ such that $x\in F$. Is every almost metrizable space $q$-space? How about $k$-spaces? A space $X$…
TXC
  • 1,332
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The difference between Urysohn's lemma and Tietze Extension Theorem.

Urysohn's lemma says that if $X$ is a normal space, then for every two disjoint closed sets $F_{1},F_{2}\in X$, there exists a continuous function $f:X\to [a,b]\in\Bbb{R}$ such that $f(F_{1})=\{a\}$ and $f(F_{2})=\{b\}$. Tietze Extension theorem…
user67803
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A condition for openness on a topology

Let $X$ be a set with topology with $\tau$. Suppose that $A \subset X$ such that for all $a \in A$ there is an open subset $U_{a}$ containing $a$ and contained in $A$. Prove that $A$ is open. I think that $A$ can be written as the union of all…
youngeAn
  • 309
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How do I show that a map $f$ is a homeomorphism iff all component maps $f_i$ are homeomorphisms?

I have the following problem: Let $I$ be a nonempty index set and let $(M_i,T_{M_i})$ and $(N_i,T_{N_i})$ be topological spaces. Moreover let $f_i:M_i\rightarrow N_i$ be maps. Finally endow $M=\prod_{i\in I} M_i$ and $N=\prod_{i\in I} N_i$ with the…
user123234
  • 2,885
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Does it make sense to talk about sub-topologies?

Upon examining the real line with the finite complement topology $\tau_{c}$, an open set $O$ is a union of certain open intervals in $\mathbb{R}$. Since $\mathbb{R}$ itself is the single interval $(-\infty,\infty)$, the empty set is the empty union…
DonjaDude
  • 55
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separation of quotient space

I found quotient space on https://en.wikipedia.org/wiki/Quotient_space, I had seen a property that I don't know how I can prove it: $X/\sim$ is a $T_1$ space if and only if every equivalence class of $\sim$ is closed in $X$.
Muniain
  • 1,453
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second countable and Hausdorff doesn't imply normal

We know that if $X$ is a second countable and regular, then it is normal. Now assume that $X$ is second countable and Hausdorff. Are there any examples to show that $X$ need not necessarily be a normal space?
Zeno cosini
  • 393
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Which topology should one use?

Consider the following problem: Which of the following statements are true about the open interval $(0,1)$ and the closed interval $[0,1]$? I. There is a continuous function from $(0,1)$ onto $[0,1]$. II. There is a continuous function …
user9464
4
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A path between subsets of $\mathbb R^n$

Let $S=\{a_i\}$ be a countable set of bounded,connected and closed subsets of $\mathbb R^n$, each of nonzero area, such that any two $a_i$ and $a_j$ may only intersect at their boundary and such that the union of all a_i cover $\mathbb R^n$ (i.e. a…
i_like_pi
  • 41
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Is topology intrinsic or is it a construction?

Given a set $M$, we can construct a topology $\tau$ to this set, which is nothing but a collection of open sets, making it a topological space $(M,\tau)$. There are many ways to do so. Continuity is defined based on the concept of open sets, i.e., a…
Jean Weigel
  • 107
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2 answers

Is my proof of $\overline{A \cup B}= \overline{A} \cup \overline{B} $ correct?

I am trying to proof $\overline{A \cup B}= \overline{A} \cup \overline{B} $, where $\overline{A}$ is the topological closure of A. I did some thinking and I came up with this. My Question is: Is my proof correct? First I am trying to proof…
John.W
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