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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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An open set containg all irrationals

Let $A$ be an open set in $\mathbb{R}$ with usual topology and $A \cup\mathbb Q=\mathbb{R}$. Does it imply $A=\mathbb{R}$ ?
Neon
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If $X$ is homeomorphic to dense subspace $Y\subseteq X$, then $X=Y$

It is a basic fact of topology that if $X$ is a topological space and $Y\subseteq X$ is homeomorphic to $X$, it does not need to occur that $X=Y$ (for example, $X=\mathbb{R}$, $Y=(0,1)$). My question is, if I add the requirement that $Y$ is dense in…
Saúl Pilatowsky-Cameo
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Is uncountable subset of separable space separable?

I have to prove that any uncountable $B\subseteq \mathbb{R}$, where $(\mathbb{R},\epsilon^1)$ is euclidean topology and topology on B is relative, is separable. And I know it's true because every subset of separable metric space is separable. But…
Meow
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A bijective function between a square and its side

Is there a simple geometric proof that there exists a continuous bijective function between a square and its side? And is there some explicit continuous function or formula $f^1(z)\mapsto (x,y)$ and $f(x,y)\mapsto z$, with $(x,y) \in…
user10903
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Topological interior of a standard $n$-simplex

A topological standard $n$-simplex is a subset $\Delta^n = \{ (x_0,x_1,...,x_n) \in \mathbb{R}^{n+1} \mid x_i \geq 0, \sum_{i = 0}^n x_i = 1 \}$ of $\mathbb{R}^{n+1}$ endowed with the subspace topology. Simplices have their own definition of…
Jxt921
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existence continuous function $f:X\to [0,1]$ such that $f(x_1)=0 , f(x_2)=\frac{1}{2}, f(x_3)=1$

Let $X$ be a Hausdorff normal topological space and $x_1, x_2, x_3$ are three distinct points.prove that there exist continuous function $f:X\to [0,1]$ such that $f(x_1)=0 , f(x_2)=\frac{1}{2}, f(x_3)=1$ I think Urysohn's Lemma will be…
poton
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example of a particular homogeneous topological space

I encountered a problem a few days ago: what is an example of a homogeneous $T_0$ space which is not $T_1$? I tried to solve this using some symmetrical properties of separation axioms which I saw before, but I couldn't. I also find this on web, but…
Mahan Mehravard
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Conditon for $f(x)$ such that $f(d(x, y))$ induces the same topology with $d(x, y)$

On Munkres' Topology, Sec 20, Q 11, for $f(x)=\frac{x}{1+x}$, $f(d(x, y))$ induces the same topology with $d(x, y)$. And I have some question here : what conditions for $f(x)$ could make same result? My attempt : If $f(x)$ satisfies $f(a+b) \leq…
random487510
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Why is the unit circle not homeomorphic to the closed unit disk?

I know that the unit circle = $\{(x,y): x^2+y^2 =1\}$ is not homeomorphic to the closed unit disk = $\{(x,y): x^2+y^2 \leq 1\}$, but I'm not sure how to prove it. I've tried with arguments with cut-points and with (path)connectedness, but still not…
hopequinn
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What is the interior of $S^n$

Is the interior of $S^n$ empty or is it all of $S^n$? My reasoning is that given any point on $S^n$, any open ball around the point cannot be contained in $S^n$, so it should be empty. However, it's all of $S^n$ according to my book.
user77134
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Continuous function between a compact set and closed set for a Tychonoff space

Please help me in proving this theorem. If $A$ is a compact subset of a Tychonoff space $X$, then for every closed set $B \subset X \setminus A$, there exists a continuous function $f:X \to I$ such that $f(x)=0$ for $x \in A$ and $f(x)=1$ for $x…
user1240174
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Question on closed sets

If $A,B\subset\mathbb{R}^n$ are closed. Let $$M=\{p\in\mathbb{R}^n\,:\,\exists\,t\in[0,1],a\in A,b\in B\;\mathrm{s.t.}\;p=t\cdot a+(1-t)\cdot b\}$$ Is $M$ closed? Why?
Anonymous999
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Given any function that maps connected sets to connected sets, is it continuous?

Suppose $f: X \to Y$ is a function with the property that any connected subset of $X$ is mapped to a connected subset of $Y$. Does it follow that $f$ is continuous?
Lukas Rollier
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Is the empty set compact in $\mathbb R$?

Here they say that the emptyset is compact. Nevertheless, in $\mathbb R$, I know that compact sets are closed and bounded. So, indeed $\varnothing $ is closed, but we have that $$\inf_{\mathbb R}\varnothing =+\infty \quad \text{and}\quad…
Bruce
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not a Hausdorff space although limits of sequences in it are unique

Prove that the co-countable topology on an uncountable set does not make it a Hausdorff space although limits of sequences in it are unique . how can I do that.I have no idea.thanks for your time.
mintu
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