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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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Can an irreducible component of a topological space be covered by the other irreducible components?

Let $X$ be a topological space, and write $X=\bigcup X_i$, where the $X_i$ are the irreducible components of $X$. Given any $X_i$, I'd like to find a point $x\in X_i$ such that $x\notin \bigcup_{j\ne i}X_j$. Such a point can always be found if $X$…
Jared
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how to prove that sigma-compact space is D-space

Please, help me. In the paper "A survey of D-spaces" by Gary Gruenhage it is written that it is easily seen that $\sigma$-compact spaces are D-spaces. Unfortunately, I don't know how to show it. The considered spaces are regular and $T_1$. Thank…
Karel Pastor
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Stone Cech compactification of the naturals a cts image of the Cantor set?

Is $\beta \mathbb{N}$, the Stone-Cech compactification of $\mathbb{N}$, a continuous image of the Cantor set?
user10
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Is Hausdorffness necessary for the classical ascoli theorem?

Munkres - topology p.278 I exactly followed the argument in the text, and I cannot find where I used hausdorffness. Where in the argument used Hausdorffness? The reason why I am asking is that the article in wikipedia requires the Hausdorffness:…
Jj-
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What is Topology of compact-convergence?

Munkres - Topology p. 283 Definition Let $(Y,d)$ be a metric space and $X$ be a topological space. Define $B_C(f,\epsilon)$ as the set $\{g\in Y^X : \sup\limits_{x\in C} \operatorname{d}(f(x),g(x)) < \epsilon \}$ for a given compact subspace $C$…
Jj-
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Why this two spaces do not homeomorphic?

Consider $\Bbb Q$ with subspace topology and $\Bbb Q\times \Bbb Q$ with product topology. Why this two spaces are not homeomorphic?($\Bbb Q$ is the rational numbers)
Aliakbar
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Can closed sets in real line be written as a union of disjoint closed intervals?

It is known that open sets in real line can be written as a countable union of disjoint open intervals. (link) I'm curious that if there is similar statements for closed sets in real line.
Gobi
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Closed unit disk homeomorphic to $\mathbb{R}^2$?

I've already shown the existence of a homeomorphism between the open unit disc and $\mathbb{R}^2$ and now I'm trying to work out whether the closed unit disc is homeomorphic to $\mathbb{R}^2$ or not. Clearly the only real difference is the fact it…
Noble.
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Continuity and image of convergent sequences

Is it true that: For a map $f:X\rightarrow Y$, between two topological spaces. If the image of every convergent sequence in $X$ is also convergent in $Y$. Then $f$ is continuous. If it is true, how to prove it? Or if it is false, what is the…
pluskid
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Is there a difference between compact and closed?

Off course there is a difference between 'compact' and 'closed', but here I am asking for something deeper. Whether a set in a topological space deserves label 'closed' is somehow depending not only on the set itself but also on its surroundings.…
drhab
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Smallest topology containing all topologies

Let $\{T_\alpha\}$ be a family of topologies on $X$. Show that there is a unique smallest topology on $X$ containing all the collections $T_\alpha$, and a unique largest topology contained in all $T_\alpha$. We can check that $\bigcap T_\alpha$ is…
Paul S.
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sequence lemma, relaxing some hypothesis of a theorem

I know that if I have a sequence $ x_n \to x,$ where $ x_n \in A,$ then $x$ is a limit point of $A.$ But the converse is not always true, at least in the case of a first countable space, if so. The question is: Is this condition of being first…
Daniel
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Zero-dimensionality of Cantor's leakier tent

Definition of Cantor's leaky tent: Let $C$ be the Cantor set in the unit interval and $p$ the point $(1/2,1/2)$ in the Euclidean plane. Let $L(c)$ the line segment in the plane connecting the point c in the Cantor set and p. Let…
LostInMath
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Prove $\bar{\mathbb{Q}}$ in $\mathbb{R}$ is $\mathbb{R}$

Take $\mathbb{R}$ in the standard (order) topology. Show that the closure of $\mathbb{Q}$ is $\mathbb{R}$. I'm new to topology, self-studying using the Munkres book. In fact I'm new to proofs. In the book he has just defined a closed set, and the…
Edward Newell
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Show: Quotient space is homeomorphic to unit sphere

An equivalence relation on $\mathbb{R}$ is given by $$ x\sim y\Leftrightarrow x-y\in\mathbb{Z}. $$ Show that the quotient space $(\mathbb{R}/{\sim},\tau_1)$ is homeomorphic to $(S^1,\tau_2)$, where $\tau_1$ is the quotient topology and $\tau_2$…
user34632
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