# 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.

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### Quotient map on S^1 such that that the quotient is an uncountable space with the indiscrete toplogy.

Based on Leinster Basic Category Theory page 135, problem 5.2.22 (b) I am trying to find a quotient map on the circle $S^1$ which results in the quotient having the indiscrete topology. Using the ...
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### The minimality of product topology.

Let $(X,\mathcal O_X), (Y, \mathcal O_Y)$ be two topological spaces, and let $\mathcal B=\{ U\times V \mid U\in \mathcal O_X, V\in \mathcal O_Y \}$. Product topology $\mathcal O$ is a family of sets ...
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### Is the surface of a 3d cube homeomorphic to the 2-sphere

Is the surface of a three dimensional cube i.e. $[0,1]^3$ (surface is like a hardboard box, I don't know if it has a symbol to represent) homeomorphic to $S^2$. If yes, is it diffeomorphic to $S^2$ ...
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### Why people usually use the term "perfectly normal Hausdorff space" instead of "perfectly normal Fréchet space"?

If a normal T2 space is a normal T1 space and a perfectly normal T2 space is a perfectly normal T1 space, then I would assume people will be more likely to use the latter term because T1 is weaker ...
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### Homeomorphism mapping half space to positive orthant

Let $P$ be the positive orthant of $\mathbb{R}^3$ and let $H := \{x_1+x_2+x_3\geq 0\}.$ Is there a way to make a (mostly smooth) map $\phi$ from $H$ to $P$ so that if $k = \{[c,c,c]^T:c\geq0\}$, then ...
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### $f$ is discontinuous. Find topological conditions that there exists a continuous $g$ st. $f(x)\geq0\iff g(x)\geq 0$.

Let $x\in X$ where $X$ is a topological space. It seems like: Claim 1. For every real-valued function $f$ on $X$, the following conditions are equivalent: there exists a continuous real-valued ...
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### $\mathbb{R}_l^2$ is not normal. [duplicate]

Let $\mathbb R_l$ be the Sorgenfrey line, i.e. $\mathbb R$ with the lower limit topology. I want to prove that $\mathbb{R}_l\times \mathbb{R}_l$ is not normal. I do not want to use Urysohn's lemma ...
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### a continous map from a topological space to itself is open

I was solving some exercises on general topology and the following question came to mind: Let $f$ be a continous map from a topological space $(X,\tau)$ to itself. Is it true that $f$ is an open map? ...
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### What separation properties are preserved by $f: X \longrightarrow Y$ continuous and surjective

I am asked prove or refute that if $X$ satisfies certain separation axiom and $f: X \longrightarrow Y$ is continuous and surjective then $Y$ also satisfies that certain axiom. For now I have found ...
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### Are there other nondegenerate examples of "non-Euclidean" metric topologies?

Note: non-Euclidean here means that the induced topology is not Euclidean, not that the metric itself is non-Euclidean (although this is also a necessary condition). Going off of literature alone, it ...
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### Can I deduce the topology on two sets by knowing the continuous maps between them?

We know that choice of topology on two set is induces a choice of which maps are continuous between two sets. Suppose we knew all the functions between two topological spaces of unknown topology which ...
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### Definition of an Open set

So from notes I'm reading, the definiton of an open set is "A subset X ⊂ (a, b) is called open in (a, b) if for every c ∈ X there is an interval (a′, b′) such that (a′, b′) ⊂ X and c ∈ (a′, b′).&...
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### Is the following set a continuum?

Let F be a non-empty family of subcontinua of a continuum X such that for any finite subfamily $F_{1},F_{2},...,F_{n}$ in F there is $C\in F$ such that $C \subset F_{1} \cap F_{2} \cap... \cap F_{n}$ ...
In their book "Homotopical Topology", Fuchs and Fomenko introduce $\mathbb{R}^\infty$ as the set of all real sequences which have only finitely many non-zero entries, i.e. $\mathbb{R}^\infty ... 0 votes 1 answer 29 views ### How to show that$A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}$is closed? I am having trouble with the following exercise: For$t > 0$consider the set $$A_t := \bigg\{u \in \mathbb{R}^n \mid u_1 = 0 \text{ and } \sum_{i=1}^n \lvert u_{i+1}-u_i \rvert \le t \bigg\}.$$ ... 1 vote 0 answers 34 views ### Is the following metric space complete? Let$X=\{x=(x_i)_{i \in \mathbb{N}} \in \mathbb{R}^{\mathbb{N}} \ \vert \ \exists N \in \mathbb{N} : x_i \geq 0 \ \ \forall i \geq N\}$and let$\bar{\rho}$be the uniform metric on$\mathbb{R}^{\...
This is a question for which I've found a number of "near-miss" results online, which may actually be answers but whose direct relevance I haven't been able to see. Say that a ring $A$ is ...