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Mathematics Stack Exchange

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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An exercise about finite intersection property in $T_1$ space

Let $X$ be a $T_1$ space. Let $\mathfrak {D}$ be a collection of subsets of $X$ that is maximal with respect to the finite intersection property. Show that there is at most one point belonging to $\bigcap_{D \in \mathfrak{D}} \bar D$ Here's my…
Metta World Peace
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How can see if something is under a topology?

X is defined as $\{a,b,c,d\}$ A topological space $X$ is called $T_{0}$ if for every pair of points $x,y \in X$, there is an open set $U$ that contains one of them and not the other. Is $XT_{0}$ is under your topology (which you must construct) …
user130306
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Compact and open subspaces using products

I'm preparing for a topology prelim, and there's one question related to compactness that I'm trying to work on. Here it is: Let $C$ be a compact subspace of $X$, and $K$ be a compact subspace of $Y$. Now let $U$ be an open set in $X \times Y$…
Libertron
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Closed Convex Subsets of $\Bbb R^2$; Find them all!

I'm sorry if I put this in the wrong area, the author has a strange habit of going on tangents. This is Question 66 in chapter 2 of Pugh's Real Analysis. Find all the closed and convex subsets of $\Bbb R^2$ up to homeomorphism. There are…
Pax
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two questions on general topology

Let $X$ be a a connected set of real numbers. If every element of $X$ is irrational then the cardinality of $X$ is Infinite Countably finite $3$ $1$ Let $(X,d)$ be a metric space and let $A⊆X$. For $x∈X$, define $$d(x,A) =…
poton
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Example of quotient restriction of quotient mapping which is not quotient

Here's the problem that I'm trying to solve: Let $f: X \to Y$ be a quotient mapping. Find an example of $f, X, Y$ where for some $A \subseteq X$, $f|_{A}:A \to f(A)$ is not a quotient mapping. If $B \subseteq Y$ is open or closed, prove that…
Matija Sreckovic
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How can one know if a set is compact?

How can one know if a set is compact? From the definition, a set is compact if for any open cover, there exist a finite subcover. However, it is not possible to list out all the open covers to a set. So is there any way one can know if it is a…
Mathematics
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Fundamental Group of a Quotient under Group Action

Let $X$ be a simply connected topological space (therefore with trivial fundamental group) and $G$ a group which acts on $X$ freely. Recently I read that for fundamental group of $X/G$ holds the equation $\pi_1(X/G) =G$. My questions are : how to…
user267839
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hausdorff space and continuous function

Here is the question I could not solve even though I have been thinking it since couple days. Let $f$ be a continuous function from $(X, \mathcal{T})$ to $(X, \mathcal{T})$. Show that if $X$ is a Hausdorff space, then $$\{x \in X : f(x)=x\}$$ …
Ridvan
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Homeomorphism of two compact sets

If I have two compact subsets, $A$ and $B$, of the plane $\mathbb{C}$, and we know that $\partial A$ and $\partial B$ are homeomorphic, can we say that $A$ and $B$ are homeomorphic?
Yee Neil
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Spaces with finite bases are compact

Let $X$ be a topological space with a finite basis $B$, and let $S$ be an open cover of $X$. Then every $U\in S$ is a finite union of members of $B$. Let $S'\subseteq S$ be a subcover such that members of $S'$ are disjoint. Then $S'$ is finite.…
Sid Caroline
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Continuous composition with one discontinuous function

Suppose $X$ is some connected topological space, $I$ is an interval and $Y$ is some topological space. Let $g: X\to I$ be a continuous and surjective function. Let $f$ be a function $I \to Y$. If $f \circ g$ is continuous, must $f$ be continuous?…
Johan
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Reasoning for definition of locally homeomorphic

I am reviewing some fundamental topological concepts, and tried to recollect what it meant for a topological space $X$ to be locally homeomorphic to another topological space $Y$. 'My Definition': I would have said that $X$ is locally homeomorphic…
azureai
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Definition of topology using separation as primitive notion

In "General Topology", chapter 1, exercise b, Kelley wrote in a note that it is possible to use the notion of separation ($A$ and $B$ are separated iff $A^k\cap B=A\cap B^k=\emptyset$) as primitive to define topological spaces and he put these three…
Daniel Kawai
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About nowhere meager and relative closed sets.

Definition: A subset $A$ of a space $X$. Then $A$ is meager in $X$ if $A=\displaystyle\bigcup_{n\in N}A_n,$ where ${\rm int}(\overline{A_n})=\emptyset$, for all $n\in N.$ And $A$ is nowhere meager in $X$, if every non-empty relatively open subset of…
Rigo
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