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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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Manifold path components are open

Can someone explain why the path components of a manifold are open? I'm a little confused at how to demonstrate this fact and it would obviously help me understand manifolds better.
user66339
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Explicit homeomorphism between quotient square and torus

Let $X = [0,1] \times [0,1]$, and let consider the quotient topology $X^* := X / ((x,0) \sim (x,1), (0,y) \sim (1,y))$. Given $r_0 > h > 0$, we define explicitly the torus as: $$ Y_{h,r_0} = \left\{(x,y,z) : z^2 = h^2 - \left(r_0 - \sqrt{x^2 +…
Clement Yung
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Sequentially compact but not compact

Let X be a subset of product {0,1}$^S$ of uncountable class of {0,1} (S is an uncountable set), consisting of all those elements for which no more than a countable number of coordinates are nonzero. (The space {0,1} here is equipped with the…
Ingrid
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If adding ONLY finitely many points to a (non-empty) subset of plane, $\mathbb{R}^2$ makes it closed, then is the closure all of $\mathbb{R}^2$?

Question: Let $E \subset \mathbb{R}^2$ be non-empty open set such that $E$'s union with finitely many points becomes a closed set. Is it necessarily true that $$ \text{closure}{\ (E)} = \mathbb{R}^2 \, \, ? $$ Equivalent Question: Looking into its…
Behnam Esmayli
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All zero dimensional spaces are completely regular.

I need to show that: All zero dimensional spaces are completely regular. Here are my definitions: Recall that a space is called zero dimensional if each point has a neighborhood base consisting of sets which are both open and closed. In particular…
Klara
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The image of first countable space

I'm trying to show that the continuous first countable image of a space does not have to be first countable, but the continuous open image of first countable space is first countable. So I let $X$ be a discrete topology so that any function defined…
Akaichan
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Question on how to show quotient topology for projection mapping

For the following question: Let $X$ be the Cartesian plane $\mathbb{R}^2$. Equip $X$ with the topology induced by the sub-basis ? $$E_{{x},{y},{\epsilon}} =\{(s,t)\in X: s=x, y-\epsilon 0$. …
Seth
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Is $f:[0,1] \to \{0,1\}$ constant?

Let $f:[0,1] \to \{0,1\}$ be continuous, where the spaces have the usual topology inherited by $\Bbb R$. Must $f$ be constant? I think it should because $[0,1]$ is connected and it can't be divided into two open disjoint sets, which should be the…
Twnk
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Real line as a product topological space

Let $\Bbb R$ be endowed with the standard topology. Clearly, we can trivially represent $$ \Bbb R \cong \Bbb R^0\times \Bbb R \tag{1} $$ and also, there is not such topological space $X$ that $\Bbb R \cong X^2$. Thus, I wonder whether…
SBF
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Inverse image of compact set under open mapping contains compact

Let $\mathbb{D}$ denote the unit disc. Say $U\subset \mathbb{D}$ is an open set such that for every $r\in [0,1)$ there exists $z\in U$ such that $|z| = r$. Given $0<\rho<1$ is it then possible to find a compact subset of $U$ denoted $K$ such that…
OgvRubin
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Minimal dense subset

Let $X$ a space; and $Y$ is a dense subset of $X$. However there may be some subset $Z$ of $Y$ such that $Z$ is also dense in $X$. Now I want to delete some elements of $Y$ which seem needless, so that I can make the new dense subset $Z$ of $Y$ be…
Paul
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Is a set with a single element like {0} dense in itself?

Self explanatory really. Couldn't find an answer. I know that there can be sets with a highest and lowest element that are dense in themselves for example $\mathbb{Q}\cap[0,1]$ but I'm not sure about sets with just a single element.
John Doe
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definition of the compact-open topology

Let $X$ and $Y$ be topological spaces and let $C(X,Y)$ denote the set of continuous maps from $X$ to $Y$. For any two subsets $A \subset X$ and $B \subset Y$ let $W(A,B) := \{ f \in C(X,Y) \mid f(A) \subset B\}$. The compact-open topology on…
Victoria Flat
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Is such space separable?

Here is a kind of space which called Katětov extension of the natural number from this remark: I have two questions on this space: 1 Is such space separable? 2 Is such space first countable? Thanks very much.
Paul
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On closure of irrational-endpoint intervals in $\mathbb{Q}$

Is $S:=\{x\in\mathbb{Q}:x<\sqrt{2}\}$ closed in $\mathbb{Q}$ when endowed with the inherited topology from $\mathbb{R}$? One argument for openness is that $S=\mathbb{Q}\cap (-\infty,\sqrt{2})$, which by definition of an open set in subspace…
Sawyer
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