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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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Countable Chain Condition for separable spaces?

I'm trying to find a proof behind a small proposition. Recall that a topological space satisfies the countable chain condition if each disjoint collection of open sets is countable. Why is it the case that every separable spaces satisfies the CCC,…
General Mustard
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Finite topologies --- what are they good for?

Do finite topologies have any practical uses other than for counterexamples
user169197
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Is any subset of an open quasi-compact set quasi-compact?

In a general topological setting what can be said about an open quasi-compact set? Is it true that a subset of such a set is compact? What if that set is open? I ask because this came up in class today with someones solution to a problem (my class…
MJoszef
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Prove: a countable intersection of open and dense sets in a compact Hausdorff $X$ is dense

Let $X$ be a compact Hausdorff space and let $\{U_n\}$ be a countable collection of subsets that are open and dense in $X$. Show that the intersection $$\bigcap\limits_{n=1}^\infty U_n$$ is dense. I tried to show that the closure of this…
Student
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A set of all rational numbers in $[0, 1]$?

I have a question that is giving me some minor grief: If $A$ is a closed set containing all rational numbers $r \in [0, 1]$, then show that $[0, 1] \subset A$. I don't really understand this question - surely that set of all values $[0, 1]$ contains…
william
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A proof that if the product of spaces is Hausdorff, each of them is Hausdorff

Is my approach to this question right? Question: Prove that if $$\prod_{\alpha \in J} X_\alpha (\neq \emptyset) $$ is Hausdorff, each $X_\alpha$ is Hausdorff. Attempt to answer: It is enough to show that if there is a $X_i,i\in J$ that is not…
user160823
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Uniqueness of limit of convergent sequence

I know that limit of convergent sequence is unique for some spaces like metric spaces, Hausdorff spaces, etc. Is there any space the limit of the convergence of sequence is not unique? -Thanks.
ruud
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Is $\mathbb{Q}$ homeomorphic to its open subintervals?

It is well known that $\mathbb{R}$ is homeomorphic to every interval $(a,b) \subset \mathbb{R}$. What can we say about $\mathbb{Q}$? Is it homeomorphic to some of its subintervals?
Crostul
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Indiscrete topology convergence

Can somebody tell me why every sequence in X converges to every point of X if we consider the indiscrete topology $\tau=${$\emptyset,X$}?
Roos Jansen
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Properties of Cp(X)

In class, we learned about the space $C_p(X)$, which is the space of all continuous real-valued functions on $X$ with the topology of point-wise convergence. To better understand the material, I started looking at problems. The one problem I found…
Maria
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Is the intersection of dense sets dense?

Recently I came across the fact that the intersection of two dense sets is a dense set. Take $\Bbb{R}$. Clearly, both $\Bbb{Q}$ and $\Bbb{R}\setminus\Bbb{Q}$ are dense in $\Bbb{R}$. However, $\Bbb{Q}\cap (\Bbb{R}\setminus\Bbb{Q})=\emptyset$. How…
freebird
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One-point compactification in non-Hausdorff spaces

I am reading Rudin's "Fourier analysis on groups" and doing a review of Topology by reading his appendix. He describes one point compactification like this: Given any topological space $S$, build $S_{\infty}=S\cup\{\infty\}$ and topologize it by…
Gadi A
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Topological properties such that if the subspace has P then the whole space also has P

Is there any topological Property $P$ if it is satisfied on a subspace topology $N$ of a topological space $X$, then it must be satisfied on a topological space $X$. Thanks.
user132983
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Every metrizable Toronto space is discrete.

$X$ is a Toronto space if for every $Y \subseteq X$ such that $|Y|=|X|$ then $Y$ is homeomorphic to $X$. I am trying to prove that every metrizable Toronto space is discrete. I have the following Ask a Topologist post which contains an overview…
SantiagoC
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Proofs involving stereographic projection

Can someone please help me finding references in which one has used following homeomorphism to solve general topology problems? $S\colon S_{0}^{n}\to \mathbb{R}^n$ is defined by for any $x=(x_1,...,x_{n+1})$, $$ S(x) =…
Abcd J
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