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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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Compact metric spaces is second countable and axiom of countable choice

Why we need axiom of countable choice to prove following theorem: every compact metric spaces is second countable? In which step it's "hidden"? Thank you for any help.
user63123
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Connectedness of a certain subset of the plane

Let $U$ be an open and connected subspace of the Euclidean plane $\mathbb{R}^2$ and $A\subseteq U$ a subspace which is homeomorphic to the closed unit interval. Is $U\setminus A$ necessarily connected?
LostInMath
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Is a perfect set a boundary?

In a topological space, is a perfect set (i.e. closed with no isolated points) always the boundary of some set?
curious
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Every infinite Hausdorff space has an infinite discrete subspace

I want to show that any infinite Hausdorff space contains an infinite discrete subspace. I am motivated by the role of $\mathbb N$ in $\mathbb R$. We know that if a Hausdorff space is finite, then it is a discrete space, but an infinite subspace of…
Anupam
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Metric is continuous, on the right track?

Let $X$ be a metric space with metric $d$. Show that $d:X\times X\rightarrow \mathbb{R}$ is continuous. The problem is taken from Munkres Topology second edition, Section 20. I know that if $d$ is a metric on $X$ then $d:X\times X\rightarrow…
user43138
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Show that the boundary of a set equals the boundary of its complement

$\newcommand{\bdy}{\operatorname{bdy}}$ I'm trying to show that $\bdy(A) = \bdy(A^c)$. I know that $\bdy(A) = \operatorname{closure} A \setminus \operatorname{int}(A) = (\operatorname{int}(A^c))^c \setminus \operatorname{int}(A)$, but I don't know…
therexists
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What is the significance of limit points?

When I had my first taste of topology a couple of years ago, our lecturer emphasized the following notions. closed set, closure, closure point open set, interior, interior point Of course, these are all basically different ways of talking about…
goblin GONE
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functions that send limit points to limit points

An exercise from Munkres states that: Suppose that $f: X \to Y$ is continuous. If $x$ is a limit point of a subset of $A$ of $X$, is it necessarily true that $f(x)$ is a limit point of $f(A)$? I think that the answer is no based on an example…
user96608
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Existence of a Minimal Cover

I'm well aware that for the sequence $x_n=\frac{1}{n}$, $\text{inf }x_n=0$ but $0 \notin (x_n)$. This made me think about something similar but when we are no longer thinking about existence of a number in a sequence but something a bit different.…
mathematics2x2life
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How to prove that $\mathbb{Q} \subset \mathbb{R}$ is not locally compact directly?

How to prove that $\mathbb{Q} \subset \mathbb{R}$ is not locally compact directly? That is, how to construct a cover of an arbitrary neighborhood (e.g. $[0, 1] \cap \mathbb{Q}$) that does not have a finite subcover?
Alexei Averchenko
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Difference of closures and closure of difference

Let $A,B$ be subsets of a topological space $X$. Is it true that $\overline{A}-\overline{B}\subseteq\overline{A-B}$? Suppose $x\in\overline{A}-\overline{B}$. So all open sets containing $X$ also contains an element of $A$. And there exists an open…
Paul S.
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Pointwise topology embedding

First let $\Lambda$ be the bijective mapping between $Y^{Z \times X}$ and $(Y^X)^Z$ defined as follows: every mapping $f: Z \times X \to Y$ defines a set of mappings from $X$ to $Y$: for each $z \in Z$ is $f_z:X \to Y$ defined as $f_z(x) = f(z,x)$.…
JT_NL
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The Disk and the Punctured Disk

Can you explane me why $$D = \operatorname{Spec}\mathbb{C}[[t]]$$ is the disk and $$D^{\times} = \operatorname{Spec}\mathbb{C}((t))$$ is the punctured disk? Or give me some links on intelligible books, lectures, etc... Thanks a lot!
Aspirin
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How is $xy=1$ closed in $\Bbb{R}^2$?

I read somewhere that $xy=1$ is a closed set in $\Bbb{R}^2$. A closed set is defined as the complement of an open set, or one which contains all its limit points. In metric spaces, it is defined as the complement of the union of balls…
user67803
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Is the number 8 special in turning a sphere inside out?

So after watching the famous video on youtube How to turn a sphere inside out I noticed that the sphere is deformed into 8 bulges in the process. Is there something special about the number 8 here? Could this be done with any number of bulges,…
Kieran Cooney
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