# 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.

38,136 questions
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### Is $\mathbb{R^n}/{\mathbb{Z}^n}$ homeomorphic to a cup?

If yes, can you explain it to me?
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### What conditions a function$f: X \to Y$Must satisfy so that the inverse image of compact is also compact?

If f is closed such that the inverse image of a unitary set is compact, I think it's worth it. But are there other conditions?
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### Nonexistence of an injective $C^1$ map between $\mathbb R^2$ and $\mathbb R$

I am getting bored waiting for the train so I'm thinking whether there can exist a $C^1$ injective map between $\mathbb{R}^2$ and $\mathbb{R}$. It seems to me that the answer is no but I can't find a ...
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### Every Topological Vector Space is Regular

Let $V$ be an abstract topological vector space over topological field $K$ (We may assume that $K = \mathbb{C}$ or $K = \mathbb{R}$ for simplicity). This means that the only think that we are allowed ...
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### Cardinality of the family of all possible dense subsets(dense in |R^n) of |R^n

Actually, for each irrational number i, the set {m+ni | m,n are integers} , is dense in |R, as there are continuum many irrationals, hence obviously there are at least continuum many dense sets . And,...
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### Is the graph of the Conway base 13 function connected?

IVT Property: If $a<b$ and $y$ is between $f(a)$ and $f(b)$, then there exists $c\in(a,b)$ such that $f(c)=y$. Theorem. Let $f:\mathbb R \to \mathbb R$ be a function with the IVT Property. If ...
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### “Every open cover admits an open locally finite refinement” - can this refinement always be realized in terms of basis sets?

Let $X$ be a paracompact topological space, let $C$ be an open cover, and let $\mathscr B$ be a basis for the topology. Does there always exist a locally finite refinement that consists of basis sets? ...
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### Density after removing a meagre subset from a non-empty $G_{\delta}$ subset.

Let $X$ be a topological space, $G\subseteq X$ be $G_{\delta}$ and $N\subseteq X$ be meagre in $X$ (and I don't know if this implies that $N$ is meagre in $G$). Question 1. Is $G\setminus N$ dense in ...
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### Does $C_c(X)$ separate points in $X$ when $X$ is a Banach space?

Suppose that $X$ is a separable, infinite dimensional Banach space. We say that a set of functions $\{f_\alpha\}_{\alpha \in A}$ separates points in $X$ if for every $x,y \in X$, there is an $\alpha$...
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### Is Top provably not cartesian closed?

It is often said that the category $\sf Top$ of topological spaces and continuous mappings is not cartesian closed. E.g., in the Wikipedia article on compactly generated spaces and in an answer on ...
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### Does classification of 1-manifolds with boundary give induced orientation of image of closed interval under a smooth immersion?

My book is An Introduction to Manifolds by Loring W. Tu. Pictured below is the last example from Section 22, Manifolds with Boundary. I have been trying to wrap my head around this for about 2 hours (...
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### Question about Topological Subspace definition

Let $(S,\tau)$ be a topological space, and let $H\subseteq S$. Using this definition, we can define the topological subspace $(H,τ_H)$ where $τ_H:=\{ U \cap H : U \in τ \}$. Now, having just begun ...
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### Find the limit point of subsets [0,1] , (0,1) , (√ 2,√ 10) , Z and (-∞,0)- [on hold]

Find the limit point of subsets [0,1] , (0,1) , (√ 2,√ 10) , Z and (-∞,0) Just help me with this
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### Is $T(C) \subseteq R^m$ closed?

Let $C \subset R^n$ be a closed, convex cone. Let $T: R^n \to R^m$ be a linear transformation. Is $T(C) \subseteq R^m$ closed? I'm very positive that the answer is NO. But I couldn't come up to a ...
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### Definition of topological space: Is Ω equal to the powerset of X?

A topological space is a set $X$ and a collection $\Omega$ of subsets of $X$ such that: $\emptyset \in \Omega$ and $X \in \Omega$ The union of any collection of $\Omega$ is in $\Omega$ The ...
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### Is $C_1 + C_2$ closed?

Two questions jump into my mind when I was working with cone, I feel they are very related to each other. one I asked here "Is $T(C) \subseteq R^m$ closed?" Second one is: If $C_1$ and $C_2$ are two ...
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### How do I calculate the Jacobian matrix of the transformation of a 1-m manifold to a chart (topology question)?

What I want to do is take a 1-m manifold (something like a circle), and transform a subset of that manifold into a chart. I want to represent that function from manifold to chart with a 1 x 1 matrix, ...
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### Are countable topological spaces second-countable?

Are countable spaces (i.e. $\mathbb{N}$ with any topology) second-countable? A countable space can have at most $2^\omega$ open subsets which suggests that a counterexample may exist. On the other ...
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### Axioms for homology and cohomology for CW complexes

This is related to assertion made in Hatcher, Algebraic Topology, Chpt 3, Sec 1 and Chpt 2. I will write down axioms for cohomology and axioms for homology is written in a similar fashion. ...
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### Obtaining mapping cone complex from $f:X\to Y$

This is related to Hatcher, Algebraic Topology Cor 3A.7(b). Cor 3A.7(b) $f:X\to Y$ induces integral homology isomoprhism iff it induces isomorphism with coefficients over $Q$ and for all $Z_p$. "...
### if $X$ is $T_1$ and limit point compact then $X$ is countably compact
A space $X$ is said to be countably compact if every countable open covering of $X$ contains a finite subcollection that covers $X$. I want to show that if $X$ is a $T_1$ space and limit point compact ...
Prove that the set $C$ of points in $\mathbb{R}^2$ such that at least a coordinate is irrational is connected set. Ok, I know that the set of Irrational numbers isn't connected but how that helps me? ...