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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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Why does paracompactness need to be not locally finite open subcollection, but refinement of opem covering?

In Munkres topology, he defines paracompactness as a generalization of compactness, as follows: $X$ is paracompact if every open cover of $X$ has locally finite open refinement that covers $X$ But why it should be "refinement" of open cover? In…
Kim Seong Hyeon
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A Hausdorff $ k$-space is minimal $ KC$ if and only if it is compact.

Atopological space is called $k$ - space if it has the property that any subset $S$ such that $ S \cap K $ is closed for all closed compact $ K $ is itself closed. The bellow theorem comes from " THE FDS-PROPERTY AND SPACES IN WHICH COMPACT SETS…
habib
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What does the quotient space of a CW complex by a subcomplex mean?

From p.8 of Hatcher's Algebraic Topology: If $(X,A)$ is a CW pair consisting of a cell complex $X$ and a subcomplex $A$, then the quotient space $X/A$ inherits a natural cell complex structure from $X$. It talks about a quotient space of $X/A$…
Daniel Donnelly
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A question about the closedness of $C$-embedded subsets

In the answer of this post it is proven that in every first countable Tychonoff space the $C^{*}$-embedded subsets are closed. Since countable pseudocharacter (i.e., points are $G_\delta$ sets) is a natural weakening of first countability, the…
Peluso
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Without Hausdorff, what implications can we prove about $k_\omega$ related to other covering properties?

In e.g. A SURVEY OF $k_\omega$-SPACES a space is said to be $k_\omega$ if it's the union of compact Hausdorff $K_n$, $n<\omega$, with a set being closed if and only if its intersection with each $K_n$ is closed. It's asserted there that $k_\omega$…
Steven Clontz
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Example of a dense open subset that is not of the form $U \cup {\sim} U$ for $U$ regular

If $U\subseteq X$ is an open set in a topological space $X$, let $\newcommand{\negation}{\mathop{\sim}}\negation U$ denote the largest open set $V$ such that $V\cap U = \varnothing$ (i.e., the set of points having a neighborhood disjoint from $U$,…
Gro-Tsen
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Difference between subspace and subset in topology

In Murkres's book Topology, I read that Y(a subset of some set X）is called the subspace of X if we consider the subspace topology. But what is the difference here between being a subspace and a subset? the definition in the book
Fitter Jack
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The Characteristic Property of Disjoint Union Spaces (Understanding theorem statement)

I am reading Introduction to Topological Manifolds by Lee, and I have a question about the statement of Theorem 3.41 (Characteristic Property of Disjoint Union Spaces) on page 64: Theorem 3.41 (Characteristic Property of Disjoint Union Spaces)…
Leonidas
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Proof that this set is open

Consider the sequence $(a_n)_n$ that is defined by $a_n:=1/n$ and the set $A:=\{a_n \ | \ n \in \mathbb{N}\} \cup \{0\}$. I want to show that $\mathbb{R} \setminus A$ is open in $\mathbb{R}$. I had two ideas to show this. $(1)$ It should be the case…
user103178981
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Are there any concrete examples of Baire spaces $X$ in which neither player has a winning strategy in BM(X)

Let $X$ be a topological space. A Banach-Mazur game $BM(X)$ is played by two players $\alpha$ and $\beta$, who select nonempty open subsets of $X$. The player $\beta$ starts a game by selecting a nonempty open subset $V_1$ of $X$. In return,…
M.Ramana
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A weaker condition of the definition of topology

Suppose that $(X,\tau)$ is a pair in which $X$ is a non-empty set and $\tau \subset 2^X$ with these conditions: (1) $X,\emptyset \in \tau$, (2) $\tau $ is closed under arbitrary unions, and (3) $U_1 \cap U_2 \neq \emptyset$ implies…
M.Ramana
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Homeomorphic infinite products of non-homeomorphic spaces

Let $X$ be be a set. Do there exist non-homeomorphic topologies $(X,\tau_1)$ and $(X, \tau_2)$ be on $X$ such that $$\prod_{i\in\mathbb{N}}(X, \tau_1)\cong \prod_{i\in\mathbb{N}}(X, \tau_2).$$ I ask because I recently learned that the infinite…
FazeZizek
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If $g∘f$ is a homemorphism, then $g$ one-one (or $f$ onto) implies that $f$ and $g$ are homeomorphisms.

I need a little explanation, please. Seymour Lipschutz - General Topology, Chapter 7, page 110: Let $X,Y,Z$ be topological spaces and let $f:X\longrightarrow Y$ and $g:Y\longrightarrow Z$ be continuous. Show that if $g\circ f:X\longrightarrow Z$ is…
Tryncha
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Continous functions on cofinite topology

Let $X$ be the cofinite topology and let $f\colon X\to X$ a non-constant function. Show that $f\colon X\to X$ is continuous iff for every inifinite subset $A\subseteq X$, $f(A)$ is infinite. I am lost with this problem and don't even know if it's…
juan_valz
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Having trouble on proving this topological continuous question in Hatcher's Notes.

I am currently reading Hatcher's notes on point set topology. I am trying to do his exercises. I am stuck on this question. 1.14 Suppose a space $X$ is the union of a collection of open sets $O_{\alpha}$. Show that a map $f:X \rightarrow Y$ is…
3j iwiojr3
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