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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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Does every finite topological space map to a family of pairwise disjoint subsets of the reals under the usual topology with closure preserved?

For a simple example, suppose $X=\{1,2,3\}$ under the partition topology $\mathcal{T}=\{\varnothing,\{1\},\{2,3\},X\}.$ The map $\mu$ taking $1$ to $\{1\}$, $2$ to $[2,3]\cap\mathbb{Q}$, and $3$ to $[2,3]\setminus\mathbb{Q}$ clearly satisfies…
mathematrucker
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Intuition behind Borsuk-Ulam Theorem

I watched the following video to get more intuition behind Borsuk-Ulam Theorem. The first part of the video was very clear for me, as I understood it considers only $R^2$ dimension and points $A$ and $B$ moving along the equator and during the video…
com
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Orbit Space question

So I'm wondering how I would show that the orbit space $I^2/D_4$ is homeomorphic to $D^2$ and $\Delta XYZ$ where X is the origin, Y is a vertex of $I^2$ and Z is the midpoint of an edge of $I^2$. I suppose I'm just confused about what I need to…
user62931
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Separable regular Hausdorff spaces have a basis of cardinality $\le 2^{\aleph_0}$? Why is $[0,1]^A$ separable iff $|A| \le 2^{\aleph_0}$?

I have a couple of questions: I need a hint to show that if X is a regular, Hausdorff and separable space, then there exists a basis $B$ of the topology of X such that $|B|\le 2^{\aleph_0}$. How do you show that $[0,1]^{A}$ is separable if and only…
J Pablo Lozano
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Prove Looped line is Hausdorff

Definition of Looped line At each point $x$ of the real line other than the origin, the basic neighborhoods of $x$ will be the usual open intervals centered at $x$. Basic neighborhoods of the origin will be the sets…
PSW
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Compact and connected set in $\mathbb{R}^2$ which is locally path-connected at only one point?

Given a compact and connected set $A$ in $\mathbb{R}^2$. Can it be locally path-connected at only one point? And can it be locally path-connected at every point except one? My thought First I thought of the topological sine curve, which is compact…
Chiquita
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Closure and Interior Comparison

I need to find a subset $A\subset \Bbb{R}$ such that the following sets are all different. $$A\qquad \mathring{A} \qquad \overline{A}\qquad \overline{\mathring{A}}\qquad \mathring{\overline{A}}\qquad \mathring{\overline{\mathring{A}}}\qquad…
Mateus Rocha
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Some properties of zero-sets and cozero-sets

Definition A subset $Y$ of a topological space is zero-set if there exist a continuous real function $f:X\rightarrow\Bbb{R}$ such that $Y=f^{-1}(\{0\})$ and so we say that a subset $Z$ of $X$ is cozero-set if $X\setminus Z$ is zero-set. So now we…
Antonio Maria Di Mauro
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Embedding $\mathbb{Q}$ with the usual topology into a power of the two point discrete space?

I came across this question and I don't have any idea where to start, so any help would be greatly appreciated! The question is as follows: "Prove that $\mathbb{Q}$ with its usual topology can be embedded into a power of the two point discrete…
katherinebarry
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A locally compact and dense subset of a Hausdorff space is open

Following a reference from "Elementos de Topología General" by Angel Tamariz and Fidel Casarrubias. Definition A topological space is locally compact if for any its point there exist a compact neighborhood. Theorem Let be $X$ a Hausdorff locally…
Antonio Maria Di Mauro
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A base for the closed sets in a topological space $ X $

A base for the closed sets in a topological space $ X $ is a family of closed sets in $ X $, such that every closed set is an intersection of some subfamily. $ \mathcal{F}$ is a base for the closed sets in $ X $ iff the familily of complements…
Curious
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Removing unnecessary sets of a refinement in a paracompact topological space.

Let $(X,\mathcal T)$ be a paracompact topological space and $\mathcal W$ be an open cover for $X$. Is there an open refinement $\mathcal U$ for $\mathcal W$, such that $$\{U\in \mathcal U\mid (\forall V\in \mathcal U)(U\not\subset V) \}$$ is a…
user59671
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Confusion with quotient topology

Let $X,Y$ be topological spaces, $\pi$ a surjective map $X\rightarrow Y$ such that $Y$ has the quotient topology induced by $\pi$. Let $I: [0,1]\rightarrow [0,1]$, $I(x)=x$. Is it true that the quotient topology on $Y\times [0,1]$ given by the map…
Lilla
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Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space.

Question: Let X be a Hausdorff space and Y be a subset of X. Then, Y with the subspace topology is a Hausdorff space. This is what I did, can someone verify this and let me know if I am correct or wrong? Also, kindly let me know if my proof need…
Math_Is_Fun
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Topology induced by metric

Get an example of a metric on a countable set that not generates the discrete topology. I think it may be a set in this way $0 \cup\{1/n:n\in\mathbb N\}$ with the metric $d(x,y)= \vert x-y \vert$ but I can not do a rigorous proof of because cannot…
Jhon Jairo
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