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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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proving that a map $f : A \rightarrow B$ is a quotient map

Suppose $A$ and $B$ are topological spaces such that $f : A \rightarrow B$ is a continuous surjective map. Assume that $\forall$ open set $U$ of $A$ its image is open. Then $f$ is a quotient map. The proof of this does not seem to bad, but I am…
Mark
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One question on subspace topology and order topology

everyone! I am learning topology with Munkres's topology book. Some examples of second chapter are very hard for me to understand. The first question is the example 3 which is on the page 90, the section of subspace topology. This example says let…
congmingniao
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Finite Product Spaces of Normal Spaces

So I am practicing some basic topolgy questions, and I came upon the statement: If $X=X_{1} \times \cdots \times X_{n}$ is normal, then each $X_{j}$ is normal. I have a proof, but I am not convinced that it is correct. Some helpful hints, better…
Rene Cabrera
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is there a simple characterisation of when compact sets are closed?

In the Euclidean spaces compact sets are always closed. This is not true for general topological spaces. Can we characterise when it is possible? Is it true for metric spaces?
Mozibur Ullah
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How do prove something is Hausdorff.

I want to prove that the set of integers are Hausdorff. Attempt : Suppose $a, b \in \mathbb{Z}$ where $a \neq b$. Then its pretty clear that if you put an open ball around each one, they are disjoint. One has to be careful though if the integers…
user133183
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Closedness and convexity of half spaces $\mathbb{R}^n$ determined by hyperplanes

Every hyperplane divides $\mathbb{R}^n$ into two "half space": the set of points "on and above" the hyperplace, $H^+ = \{ \mathbf{x} \mid \mathbf{a} \cdot \mathbf{x} \geq \alpha \}$, and the set of points "on and below" the hyperplace, $H^- = \{…
Chun-Kai
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Showing the quotient map is open

If I have a topological space $X$ and a subgroup $G$ of $\operatorname{Homeo}(X)$. Then defining an equivalence relation $x \sim y$ iff there is a $g\in G$ s.t. $g(x) = y$. I'm trying to show that the quotient map $q: X \to X/R$ is open. I can just…
Wooster
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The continuity of a function in the uniform topology

I am doing the exercises in the book Topology(2nd edition) by Munkres. Here is my question(page 127, question 4(a)): Let $h:R\to R^\omega$ be a function defined by $h(t)=(t, t/2, t/3, \ldots)$ where $R^\omega$ is in the uniform topology. Is $h$…
math 101
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Is the set of functions from a compact to complete space complete?

I've noticed that it's often interesting to consider what types of properties of a topology are preserved or lost when mapping between different types of spaces. This led me to wonder about spaces of functions themselves. In this case, suppose I…
Gotye
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A question about the restriction of quotient maps to subsets of domain.

Munkres' "Topology" (Second edition) says the following: Let $p:X\to Y$ be a quotient map; let $A$ be a subspace of $X$ that is saturated with respect to $p$; let $q:A\to p(A)$ be the map obtained by restricting $p$. If $A$ is either open or closed…
user67803
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Properties of the cofinite topology on an uncountable set

Let $X$ be an uncountable set and let $\mathcal T = \{U \subseteq X : U = \varnothing\text{ or }U^c \text{ is finite} \}$. Then is topological space $(X,\mathcal T)$ separable? Hausdorff? second-countable (has a countable…
Struggler
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How does this metric induce the product topology?

After reading about the remarkable result of Tychonoff's theorem, I've been going through some exercises to better understand the product topology. At the end of the day, this one has eluded me. Perhaps someone could shed some light on why the…
Gotye
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Is the surface of the body/skin homeomorphic to a torus?

This may seem like an odd question, but I'm asking myself if the surface of the body/skin can be made into a torus (and how it would look like, e.g. the positions of the legs, arm asf). My intuitive answer is yes, but since I'm not that deep into…
holistic
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How can I show this set of sequences is compact?

I've been mulling this idea over for a while now with little progress. Maybe someone here will have more success with it. Here, let $\ell_1$ be the set of real-valued sequences $\{x_n\}$ such that $\sum_{n\in\mathbb{N}}|x_n|\lt\infty$, and let…
Gotye
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A completely regular space is a Hausdorff space

A completely regular space is a $T_1$ space $X$ with the property that if $x\in X$ and $F$ is any closed subspace of $X$ which does not contain $x$ then there exists a function $f\in\mathcal{C}(X,\mathbb{R})$, such that $f(x)=0$ and $f(F)=1$. (Here…
QED
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