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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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(Alexander Horned Sphere) A counterexample to the possible generalization of the Schonflies Theorem

First of all: The Schonflies Theorem: Let $f : S^1 \rightarrow \mathbb R^2$ be an embedding. Then there exists a homeomorphism $F : \mathbb R^2 \rightarrow \mathbb R^2$ such that $F_{|S^1} = f$. And also according to the same book: But there…
user200918
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Describe the interior of Cantor set

Describe the interior of Cantor set I think the interior is empty because the cantor set of nowhere dense, but as I write it correctly?
Chalie Her
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Homeomorphism between $\mathbb{Q}$ and $\mathbb{Q}(>0)$, and $\mathbb{Q}(\ge 0)$

I want to ask about the homeomorphism between $\mathbb{Q}$, $\mathbb{Q}_{>0}$: the rationals greater than $0$, and $\mathbb{Q}_{\geqslant 0}$: the rationals $\geqslant 0$? For $\mathbb{Q}$ and $\mathbb{Q}_{>0}$, I can use the back and forth map to…
Long
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What does compactness actually mean

You're probably already thinking "silly person" but hear me out. Compactness: Every open cover (of a set $A$) has a finite subcover (this means every cover $\{U_\alpha\}_{\alpha\in I}$ where $A\subset\bigcup_{\alpha\in I}U_\alpha$ or…
Alec Teal
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Understanding the Zariski topology on $\mathbb{R}$

I'm having trouble understanding the concept of the Zariski topology on $\mathbb{R}$. My notes say that subsets of $\mathbb{R}$ are closed iff they consist of finitely many points or if they are all of $\mathbb{R}$. So does that mean no intervals…
user26069
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Munkres Topology Question article 17 problem 5

Let $X$ be an ordered set with the order topology. Show that $\overline {(a,b)}\subset [a,b]$.Under what conditions does equality hold? Proof:Since $\overline{(a,b)}$ is the smallest closed set containing $(a,b)$ and $[a,b]$ is a closed containing…
Learnmore
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How can I find a homeomorphism from $\mathbb{R}^n$ to the open unit ball centered at 0?

I'm trying to prove that the open ball of radius 1 centered at the origin in $\mathbb{R}^n$ is homeomorphic to $\mathbb{R}^n$. I believe the "shrinking map" from $\mathbb{R}^n$ to the ball given by $x \mapsto \dfrac{x}{1 + |x|}$ does the job, but…
Axesilo
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Proving a function is an open map, with limitations

Following from a previous question I posted in here, given the same thing, i.e. an open set $U$ in $\mathbb{R}^2$ and a continuous function $f:U\rightarrow \mathbb{R}^2$ such that for each $u\in U$ exists a neighbourhood $V_u$ in $U$ such that…
Eric_
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Product Topology of Discrete Sets

If I am examining two sets $X$ and $Y$, each with the discrete topology, will $X\times Y$ have a discrete topology? My understanding is yes. I believe this because $X\times Y$ is the finite product of discrete spaces. Every point in $X$ is open and…
Jwills
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A question on the proof of 14 distinct sets can be formed by complementation and closure

In Munkres, problem 20 of Section 2-6, it says that 14 distinct sets can be formed by complementation and closure. I see only five so far. Let f be the function of closure mapping and g be the function of complementation mapping. It is clear, f,g,…
user45765
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Is continuous image of dense set dense?

Here is the problem: Let $f : X \to Y$ be a continuous map between topological spaces. Let $E$ be a dense subset of $X$ (that is, $\operatorname{Cl} E = X$), where $\operatorname{Cl}E$ represents the closure of $E$. Prove: $\operatorname{Cl} f(E)…
Allen Cho
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Homeomorphism between compactification of real line and unit circle.

I'm currently working on a topology assignment which is, unfortunately, due today. As part of that, I need to show that one-point compactification of the real line, $\mathbb{R}\cup\infty$ is homeomorphic to the unit circle. I've come so far to have…
Tom
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Can a disconnected set be simply connected?

My calculus professor told me today that a set consisting of two simply connected regions that are disconnected from each other (say, two disks, each of radius $\frac 12$, centered at $(-1, 0)$ and $(1, 0)$ respectively) is still considered simply…
Joe Z.
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The Line with two origins

I have seen descriptions of the "line with two origins" using quotient spaces. My professor has defined it in an alternate way. However, I can not wrap my head around how the following descriptions forms a line with two origins. Consider…
user7090
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Locally compact subspace is an intersection of an open and closed set

Let X be a locally compact topological space. I need to prove that if $M\subset X$ is a locally compact subspace of X then there exist $U,F\subset X$ such that U is open and F is closed, and $M=U\cap F$. It can be assumed that for every open $x\in…
giladude
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