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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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Could the closed discrete subset of the first countable regular space have arbitrary small closed nbhd?

Suppose $X$ is any first countable regular space, and $D$ is the countable closed discrete subset of $X$, then could $D$ have arbitary small closed nbhd, i.e., for any open set $U$ of $X$ which contains $D$, $D$ has an small open set $V$, which…
Paul
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Topologies on a 3 point space

Let $Y_1$ denote a 2-point discrete space, $Y_2$ a 2-point indiscrete space and $Y_3$ a 2-point space that is neither discrete nor indiscrete. Let $X$ denote a set with 3 points and describe topologies on $X$ such that $X$ has subspaces homeomorphic…
user62931
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The Line with Uncountably Many Origins: Second Countable?

Motivation: The line with two origins can be constructed via taking two copies of ${\mathbb R}$ and identifying $x$ on the first copy with $x$ on the second copy for each $x\neq 0$. We note that because this space is separable and first countable,…
user2959
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Uniqueness of a continuous extension of a function into a Hausdorff space

Suppose that $A\subset X$ and suppose that $f : A \to Y$ is a continuous function with $Y$ Hausdorff. Show that there is at most one continuous function $g : \bar{A} \to Y$. My try: Suppose there are two extension $g$ and $h$, then for some…
user62532
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Proof that the interval and the plane is not homeomorphic

I tried to prove this by contradiction. I assumed that there is a homeomorphism $f: (0,1) \longrightarrow \mathbb{R^2}$. A submap $ab(f): (0,1)\setminus\{\frac{1}{2}\} \longrightarrow \mathbb{R^2}\setminus\{f(\frac{1}{2})\} $ (it's also…
Oiale
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Closure, interior and boundary point

Prove that a point belongs to $A^{-}$ if and only if it is either an interior or a boundary point of $A$; where $A^-$ is the closure. To be an interior point: A point $x\in \mathbb{R}$ is the interior point of a set $A\subset \mathbb{R}$ if there…
Lays
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Can a non-perfect-Polish topology have the same "sequential convergence structure" as a perfect Polish topology?

Given a topological space $X$, by its sequential convergence structure I mean the full information about the convergent sequences in $X$ together with their limits. (I guess to be formal, it is the subset of $X^{\mathbb{N}} \times X$ consisting of…
stochastic randomness
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Intersection of closed and open set

In arbitrary topological space, let $A$ be open and $B$ be closed. If $\text{int}(B) \neq \emptyset$ and $A \cap B \neq \emptyset$, is it guaranteed that $A \cap \text{int}(B) \neq \emptyset$? If not, is there any reasonably weak additional…
Michał Zapała
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Proof that the Michael Line is Hausdorff ($T_{2}$)

Can I have feedback on my proof, please? Prove the Michael line topology, $T_\mathbb{M}=\{U \cup F: U$ is open in $\mathbb{R}$ and $F\subset \mathbb{R}\setminus \mathbb{Q}\}$ is $T_{2}$ (Hausdorff). Let $a,b \in \mathbb{R}$. WLOG, let $a
PerpetualStudent
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question about bases of a given topology

Given a set $X$ with a given topology $T$, can $T$ have more than one basis that generates $T$? Can you explain your answer? (I don't think that it can, but I can't think of why not either).
user39794
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Show $\{-n + 1/n \ : n \in \mathbb{N}\}$ is a closed set

I have the set $$A = \{-n + 1/n \ : n \in \mathbb{N}\}$$ My attempt I tried to find some limit point in A, but $$ \lim_n (-n + 1/n) = -\infty $$ Is there anyone to help?
Moreblue
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Definition of Disjoint Union Spaces

The following is the definition of disjoint union spaces in John Lee's "Introduction to Topological Manifolds": Let $(X_{\alpha})_{\alpha\in A}$ be an indexed family of nonempty topological spaces. We define the disjoint union topology on…
Koda
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Closure of open set in a dense subspace of topological space.

Let $Y$ is a dense subspace of topological space $X$ and $U \mathop \subset \limits^{open} Y$. Say $U=V\cap Y$ with $V$ open in $X$. My purpose is to show that $Cl_{Y}(U)= Cl_{X}(V)\cap Y$. and i have two Question; Is it always true that…
TXC
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The product topology on $\mathbb R_d\times \mathbb R$ vs. the dictionary order topology on $\mathbb R\times \mathbb R$

Show that the dictionary order topology on $\mathbb R\times \mathbb R$ is the same as the product topology $\mathbb R_d\times \mathbb R$ where $\mathbb R_d$ is $\mathbb R$ with the discrete topology. My thoughts: First, the space $\mathbb R_d\times…
user557
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Intersection of Compact Sets Is Not Compact

What is an example of a topological space $X$ such that $C,K\subseteq X$; $C$ is closed; $K$ is compact; and $C\cap K$ is not compact? I know that $X$ can be neither Hausdorff nor finite. I am interested in this question because I recently read the…
wjmolina
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