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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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Topology of pointwise convergence and point-open topology are equivalent

Definition VII.1. In Nagata's Modern General Topology construct Topology of pointwise convergence as follows: Given $x_1,\cdots,x_n\in X$ and $O_1,\cdots, O_n\in \mathcal{T}_Y$, let $$[x_1,\cdots,x_n;O_1,\cdots, O_n]=\{f\in C(X,Y): f(x_i)\in…
TXC
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Does my set have empty interior?

I am trying to figure out if my set has empty interior or not. It is defined as follows. Let $(q_n)_{n \in \mathbb{N}} = \mathbb{Q}^2$, we define $U = \bigcup_{n \in \mathbb{N}} B_{\frac{1}{2^n}}(q_n) \subset \mathbb{R}^2$ and $A = U^C$. Now we fix…
Linrod Míriel
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meaning of topology on a finite set

I have just started learning topology on my own and it's been quite intuitive and well-motivated while it was defined on the set of real numbers. However, in many books authors, for the sake of simplicity, usually begin demonstrating a topology on…
Maulsharif Aidos
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$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then

$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then $0$ is an isolated point $(-2,0]$ is an open set $0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$ $(-2,0)$ is open set. $0$ is not…
Myshkin
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Continuity of a function of product spaces

Let $f: X \times Y \rightarrow Z $ such that. $ \forall x_0 \in X , f_{x_0} : Y \rightarrow Z, y\mapsto f(x_0,y)$ is continuous. $ \forall y_0 \in Y , f_{y_0} : X \rightarrow Z, x \mapsto f(x,y_0)$ is continuous. How i can prove that $f$ is…
user73577
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Is every projection continuous?

Let $X$ be a topological space and $f: X^2\to X$ be a projection onto the first factor. Is $f$ continuous? Thanks for your help.
Paul
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Do infinitely many points on earth have the same temperature as their antipodal?

Let $X=S^2$ be the unit sphere in $\mathbb{R}^3$ and $T:X\rightarrow \mathbb{R}$ be a continuous function. My topology textbook claims that the set $A=\{x \in X\ |\ T(x)=T(-x)\}$ has an infinite number of elements. The fact that $A$ is non empty is…
Roberto Faedda
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Is a compact, connected subset of $\Bbb{R}^n$ whose boundary has empty interior inside it determined by its boundary?

Suppose $A_1,A_2$ are bounded, closed, connected subsets of $\Bbb{R}^n$, such that $\partial A_i$ has empty interior inside $A_i$ (for both $i$). Is it true that if $\partial A_1=\partial A_2$ then $A_1=A_2$? This question is inspired by this one,…
Cronus
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Give an example of a (path-connected) covering space which is not a regular covering space.

I am having great difficulty with the following qualifying exam problem and would appreciate some help. Thank you so much in advance. Give an example of a (path-connected) covering space which is not a regular covering space.
kyle
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$\mathbb{Q}$ is not open, is not closed, but is the countable union of closed sets.

I want to prove that $\mathbb{Q}$ (the set of rational numbers) is not open, is not closed, but is the countable union of closed sets. I tried to show that $\mathbb{Q}$ doesn't contain all of its limit points which would imply not closed. However, I…
monalisa
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Can a closed interval be an open set in a topological space?

In the exercises of the book 'Topology without tears', we are asked to prove that the following collection of subsets of $\mathbb{R}$ is a topology: The set containing $\emptyset$, $\mathbb{R}$, and every closed interval $[-n,n]$, for $n$ any…
Mouli
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Showing $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$

Let $\mathbb{Q}$ be the set of rationals, as relative topology to $\mathbb{R}$. How to show that $\mathbb{Q}$ is homeomorphic to $\mathbb{Q}^2$? More general, if $T$ is a countable dense space(no isolated poiont) with every singleton closed, how to…
Leitingok
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Construction of Basis for Product Topology

In a book I read the following: Let $(E_i, \mathcal{T}_i)_{i\in I}$ be a family of topological spaces, and let $E = \prod_{i\in I} E_i$ be the Cartesian product of the $E_i$'s. Denote by $\pi_i$ the natural projection from $E$ into $E_i$, defined…
StefanH
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Is topology just the minimal mathematical structure you need to define on a set just to define continuous functions?

I looked for a lot of explanations for the idea of defining a topology for a set to make a topological space, I found a lot of clever explanations but most of them seem like just "concrete explanations that may seem like the original idea but…
Ziad H. Muhammad
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Practical meaning of nowhere dense sets

What is the practical meaning of a subset being nowhere dense on another set? What does it mean apart from the definition (its closure to have no interior points)?
Catherine
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