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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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Can $\overline{Y}$ have non-empty interior if $Y$ has empty interior

(below, $\overline{Y}$ denotes the closure of $Y$) Given a metric space $X$ let us define a subset $Y$ to be nowhere-dense if and only if $\overline{Y}$ has empty interior. It is obvious that if $\overline{Y}$ has empty interior, then so does $Y$,…
Étienne Bézout
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Is the riemann sphere compact even though the complex plane isn't?

The Complex plane, set of all $z=x+iy$ where $x$ and $y$ are real, surface area equals cross product of $x$ and $y$ equals aleph-something (that's not the question). Projecting the plane onto a sphere, and adding the complex infinity to the set…
DarthPickley
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Zariski topology over $\mathbb R$

What is a "Zariski topology on $\mathbb R$"? I don't think I quite understand the definition of a "Zariski topology". Thank you.
Gerald
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Long ray: Proof of being locally euclidean

Consider the topological space $X=\omega_1 \times [0,1)\setminus (0,0)$ equipped with the order topology that arises from the lexicographical order. I want to show that this space is locally euclidean. Clearly, for points $(\alpha,x)$ with $x>0$ I…
principal-ideal-domain
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Shrinking Lemma for Arbitrary Open Covers of Normal Spaces

I was reading Munkres' Topology book and I came across this Shrinking Lemma: If {$U_1, ..., U_n$} is an open cover of a normal space $X$, then there is an open cover {$V_1, ..., V_n$} such that the closure $\overline{V_i} \subset U_i$ for each $i =…
Ryan Tran
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Why is $(.5, 1]$ considered an open set in $[0, 1]$?

Why is $(.5, 1]$ considered an open set in $[0, 1]$? This is from a topology textbook.
Jeff
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Intersection of regularly open sets is regularly open

This is an exercise in Willard's General Topology. A subset $B$ of a topological space is called regularly open iff $Int(Cl(B))=B$. I need to show that if $U$ and $V$ are regularly open then $Int(Cl(U\cap V))=U\cap V$. I've been using the facts…
Weltschmerz
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Let $X$ be an infinite set with a topology $T$, such that every infinite subset of $X$ is closed. Prove that $T$ is the discrete topology.

Let $X$ be an infinite set with a topology $T$, such that every infinite subset of $X$ is closed. Prove that $T$ is the discrete topology. I have somewhat of an answer but I don't think it's enough to prove it, especially with respect to the…
ZZS14
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Quotient of unit interval homeomorphic to unit circle

Let $I=[0,1]$ be a subset of $\mathbb{R}$ (with standard topology). Define the equivalence relation $x\sim x'$ iff $x=x'$, or $x=0$ and $x'=1$, or $x=1$ and $x'=0$. Show that the set of equivalence classes $I/{\sim}$ (with quotient topology) is…
Adam
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Homeomorphism between punctured plane and cylinder

I am asked to prove that the cylinder and the punctured plane are homeomorphic. I understand that I need to find a function that maps every point in the plane to a point on the cylinder. I can represent every point on the plane in polar…
user7090
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Countable topological basis

Let $(E,\mathcal{T})$ be a topological space with a countable basis (a second countable space). If $\mathcal{U}$ is a topology on $E$ with $\mathcal{U} \subset \mathcal{T}\,$ ($\mathcal{U}$ is coarser than $\mathcal{T}$), does $(E,\mathcal{U})$ have…
francis-jamet
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Is real line disconnected in discrete topology?

My professor told me that the real line is connected in any topology. But i am thinking that if for example we consider the discrete topology in $\mathbf R$ i.e every subset is open then for any $x$ in $\mathbf R$ $(-\infty,x) \cup [x,\infty)$ would…
lukanikos
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Verifying that these sets form a topology

I am solving Exercise 4.1, Question 17(v) from Topology without Tears (link) by Sidney Morris. (This exercise is marked with a star.) Let $S = \{ \frac{1}{n} \,:\, n \in \mathbb N \}$. Define a set $C \subseteq \mathbb R$ to be closed if $C = A…
Srivatsan
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Is there anywhere we use a fibration which is not a fiber bundle

What I currently meet are all fiber bundles.
hao
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Characterization of a closed mapping

I'm having troubles with this basic question. I think it is really easy but I feel like blocked or blind with it. Suppose we have a mapping $f\colon X \to Y$, the following are equivalent: a) $f$ is closed, b) $\forall U$ open on $X$, $\{y\in Y\mid…
Nelson
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