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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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Every open set has a proper open subset. What spaces satisfy this property?

Suppose $X$ is a topological space satisfying the following property: every (nonempty) open set of $X$ has a (nonempty) proper open subset. Does this property have a name? What are some spaces with this property? What are some properties that imply…
étale-cohomology
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$\mathbb R $ is uncountable

The uncountability of $\mathbb R $ can be proved by two beautiful methods.. One is by proving the sequence of 0and 1 are uncountable using Cantor's diagonal process in which we choose any countable subset of the set of all sequence of 0 and 1. And…
Samiron Parui
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Are all proper maps continuous

Some textbooks define proper maps as continuous maps which have inverse images of compact sets compact. Whereas wikipedia defines them to be having inverse images of compact sets compact. Wikipedia definition does not include continuity? Does…
math31
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Clopen subsets of $A^\Bbb N$ for finite $A$

Let $A$ be a finite set with the discrete topology and let $X = A^\Bbb N$ be the product space. Let $$ \pi_n:X\to A^n $$ be the projection map, i.e. $\pi_n(x_1,\dots,x_{n},x_{n+1},\dots) = (x_1,\dots,x_n)$. Each such map induces a partition of…
S.D.
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For any set $X$, is there a Hausdorff $Y$, and a disjoint family $\{Y_x:x\in{X}\}$ of dense subsets?

Greets I want to prove the following: For any set $X$ there exists a Hausdorff space $Y$ and a family $\{Y_x:x\in{X}\}$ of disjoint subsets of $Y$, each dense in $Y$. I want to prove this assertion since by proving it, I can use it to prove that any…
Camilo Arosemena-Serrato
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If $X$ is a topological separable space then every open set is a union of a countable number of open balls?

I am studying some real analysis (I am in the part before defining Borel's $\sigma$-algebra) and there is a reminder in my lecture notes that states that if $X$ is a separable metric space then every open set is a union of a countable number of open…
Belgi
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Properties that are not preserved under homeomorphism

Homeomorphism establishes a very strong relationship between topological spaces. We know that many important properties such as compactness and connectedness are preserved under homeomorphism, and fundamental groups of such spaces are isomorphic. I…
Yoni
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A metrizable Lindelöf space has a countable basis

$X$ is called metrizable Lindelöf space if $X$ is a metrizable space and every open covering of $X$ contains a countable subcovering. Would you help me prove that $X$ has a countable basis? Thanks
beginner
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Definition of irreducible components in topological spaces

First of all the definition of irreducibility: A topological space $X$ is called reducible if $X$ can be written as a union of two non-empty and closed proper subsets of $X$. We call $X$ irreducible if it is not reducible. A subset $F$ of $X$ is…
Diglett
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Is the intersection of an open set with a closed set open, closed, or neither?

I was trying to determine whether the following set is open or not: $$C:=\{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z > 3 , z \ge -1 \} .$$ To do so, I tried to show that the following two other sets (whose intersection, $C_1 \cap C_2$, equals $C$)…
justdoit
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Why is Completeness and Compactness not equivalent in Normed Spaces?

Given a complete normed space $X=(X,\|\cdot\|)$. Every Cauchy sequence converges in it. I am not able to understand why we can't show that every bounded sequence in $X$ will have a convergent subsequence. Please give an example to clarify why…
ajay pawar
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Linear transformation maps the first quadrant to a closed set

My question is Let $A: \mathbb{R}^n \to \mathbb{R}^m$ be a linear transformation, how can one prove that $\{Ay|y\geq 0\}$ is a closed set? Here $y\geq 0$ means each component of $y$ $\geq 0$. I can simplify the question in two ways Since the…
chan kifung
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Embedding $RP^2$ into $R^4$

I have a homework Question which asks to show that the map \begin{equation}f:R^3\rightarrow R^4, f(x,y,z)=(x^2-y^2,xy,xz,yz)\end{equation} Induces an embedding of $RP^2$ into $R^4$ Overall I have a fairly good idea of how I want to go about showing…
ai.jennetta
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Set of limit points is closed in a general topological space

Let $(X, \mathcal{T})$ be a topological space. I came across a question asking for the proof of the fact that the set of limit points $S'$ of any subset $S \subseteq X$ is closed assuming the space is Hausdorff. However, is this always true for a…
user334263
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How are a set of matrices a topological space?

For instance, is the topology on the set of all $2 \times 2$ real matrices basically $\mathbb{R}^4$
Smith
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