Mathematics Stack Exchange
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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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vector field on sphere with two handles

Consider a sphere with two handles. If I don't make a mistake it is torus with one handle: Can you give me an example of vector field on it with one singular point. Thanks.
Aspirin
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A space that is not homeomorphic to itself

I read the following sentence in Wikipedia. It is the second one in the paragraph. ... a $T_0$ topological space which is homeomorphic to itself and exhibits pointwise convergence... Isn't every space homeomorphic to itself? The identity mapping…
superAnnoyingUser
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Intersection of Dense Sets

For my Topology final yesterday, I proved that the intersection of two open dense sets is again a dense set. I used the fact that a set $A$ being dense in a space $X$ is equivalent to $X - A$ having empty interior. My proof, however, only seemed to…
Thomas Credeur
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bijective continuous function on $\mathbb R^n$ not homeomorphism?

Suppose we have a bijective continuous map $\mathbb{R}^n\to\mathbb{R}^n$ (relative to the standard topology). Must this map be a homeomorphism? I have little doubt about this. I think that if it happens, I guess it's true, I've heard it is true, but…
Daniel
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Functions from the space of subsets of a set to the space of topologies

Let $X$ be a set and $\mathcal P \left({X}\right)$ be the set of subsets ordered with inclusion. Let $\mathcal{T}(X)$ be the set of all topologies on $X$ ordered with set inclusion. Let $\mathcal{T}(X)/ \sim$ denote the quotient set that identifies…
Saal Hardali
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Covering of a topological group is a topological group

If we have a covering $p:H\rightarrow G$ where $G$ is a topological group, then $H$ is also a topological group. The multiplication function can be defined as follows. Consider the map $f:H\times H \rightarrow G$ which is a composition of the map…
Down
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Is a space where only finite subsets are compact sets always discrete?

If in a topological space only finite subsets are compact sets, is it then the discrete topological space? Thank you.
Myshkin
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Extending injection in Hausdorff spaces

Let $f : X \to Y$ be a continous locally injective map. Then, if $f$ is injective in a compact subset, it is injective in a neighborhood of that compact. This result was proved here when $X, Y$ are $\mathbb{R}^{n}$ . As far as I see, the result…
Zanzag
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How big can a quotient space be?

Let $X$ be a space of weight $w(X)=\kappa$. Suppose $q:X\rightarrow Y$ is a quotient map. If $q$ is open, or if $Y$ is compact, then $w(Y)\leq w(X)$. In general it is possible for $w(Y)$ to be larger than $w(X)$ (consider…
Tyrone
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Topology on power set such that union is continuous

Let $C$ be a set and let $2^C$ be its power set. Consider the union function $\cup: 2^C\times 2^C\rightarrow 2^C$ such that $\cup (X,Y)=X\cup Y$. For a fixed $Y\in 2^C$, let $\cup_Y$ be the evaluation map $\cup_Y(X)=X\cup Y$. Are there natural…
Math-Phys-Cat Group
  • 511
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2 answers

countable family of open sets

Is there a countable family of open subsets of ${\bf R}$ or $[0,1]$ such that each rational belongs to only finitely many of the open sets and each irrational belongs to infinitely many of the sets?
user558840
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Can different topological spaces have the same square?

For topological spaces $X$ and $Y$, is it possible that $X \times X$ and $Y \times Y$ are homeomorphic, but $X$ and $Y$ are not homeomorphic? (I poked around with finite spaces, and manifolds, and the Cantor set, without seeing any examples.) This…
Hew Wolff
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Does $A$ open, non-empty, $\frac {A+A} 2=A+1$ imply $A=\mathbb R$?

This is elementary and it may have an easy proof. Does $A$ open,non-empty in $\mathbb R$, $\frac {A+A} 2=A+1$ imply $A=\mathbb R$? There are many sets satisfying this equation: $\mathbb Q$, dyadic rationals etc. I can show that if $A$ satisfies this…
Kavi Rama Murthy
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Prove that there is a unique topology given interior operator

Suppose $f: \mathcal{P}(X) \to \mathcal{P}(X)$ is a function that satisfies, for every set $A,B \subseteq X$ $(I_1): f(X) = X$ $(I_2): f(A) \subseteq A$ $(I_3): f(A \cap B) =f(A) \cap f(B)$ $(I_4): f(f(A)) = f(A)$ Prove that there exists a unique…
user370967
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3 answers

Is there exist a homeomorphism between either pair of $(0,1),(0,1],[0,1]$

As the topic is there exist a homeomorphism between either pair of $(0, 1),(0,1],[0,1]$
Mathematics
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