# Tagged Questions

The compactness tag is for questions about compactness and its many variants (e.g. sequential compactness, countable compactness) as well locally compact spaces; compactifications (e.g. one-point, Stone-Čech) and other topics closely related to compactness.

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### defining a topology by its compact sets

The goal. Let $X$ be a set endowed with Hausdorff topologies $\tau_w$ and $\tau_n$, such that $\tau_w\subseteq\tau_n$. Let $\mathscr{C}$ denote a family of subsets $A\subseteq X$, which satisfies ...
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### An example of a compact multiplicatively unbounded ring

My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
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### When is an Open Set Homeomorphic to the Interior of its Closure?

Let $X$ be a topological space and $U \subseteq X$ open. Then $U \subseteq \operatorname{int}(\operatorname{cl}(U))$. I am looking for known assumptions on $X$ and $U$ such that one of the following ...
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### Regularly open, co-zero sets in compact Hausdorff spaces

It follows from the definition of a completely regular space that such spaces have a base consisting of co-zero sets, that is, sets whose complement is the zero set of some real-valued, continuous ...
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### Elementary proof of compact space = exhaustible space?

(This is a repost of a question I asked last year on cs.stackexchange.) The work of Martín Escardó has demonstrated close parallels between classical topology on one hand and computability on the ...
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### Group orderable iff all its finitely-generated subgroups are orderable

I want to proof this specifically using the Compactness Theorem from propositional logic (this is an exercise from Model Theory, Hodges). $G$ orderable means there is a total ordering s. t. $g\leq h$ ...
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### Is $X$ pseudocompact

The following example with a little modified from the handbook of set theoretic topology, Page 574: Let $\kappa$ be any cardinal for which there exists a family $\{H_\alpha: \alpha < \kappa\}$ ...
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### Is $X\simeq [0,1]$?

Suppose that $X$ is a metric continuum irreducible between two points $p$ and $q$. Suppose further that whenever $U$ is a connected open set missing $p$ and $q$, we have $X\setminus U$ has two ...
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### Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.

Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is compact. The following is my proof I'd like to know if it is correct. Proof: I will use the fact that ...
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### Prove that every pseudocompact metric space is compact

This is from Real Mathematical Analysis by Pugh, problem 2.85(a). I've seen proofs but they've used concepts that haven't been covered up to this point, like the Tietze extension theorem, metrizable ...
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It seems like $\beta\mathbb R$ has been heavily studied, but I am interested in learning more about $\beta(\mathbb R ^\omega /u)$. $\mathbb R ^\omega /u$ is a proper extension of $\mathbb R$ when $... 0answers 108 views ### Lie groups with structure constant$f_{abc} \neq f_{bca}$. The structure constant$f_{abc}$of Lie group is defined by the commutators of generators, $$[T^a,T^b]=i f_{abc}T_c$$ automatically$f_{abc}=-f_{bac}$. Can someone give a list of explicit examples ... 0answers 690 views ### What is compensated compactness? As the title says, what is compensated compactness? I see people talk about it in the books and papers I am reading but I can only find hand wavy definitions when I look online. Is there a definition ... 0answers 175 views ### Prokhorov theorem in locally compact Hausdorff space? Prokhorov theorem gives a compactness condition in the space of probability measures on a Polish space. I am wondering whether we have similar conditions for probability measures on more general ... 0answers 201 views ### Is this proof correct about compact sets inside open sets? I've been solving the following problem: "If$U\subset\mathbb{R}^n$is open and$C\subset U$is compact, show that there is a compact set$D$such that$C\subset \operatorname{int}(D)$and$D\subset U$... 0answers 375 views ### Every Borel set is the union of an increasing sequence of Bounded Borel sets? I am currently working with the book by Halmos, and i can't quite get past this one. It states that: "Every Borel set can be written as an increasing sequence of Bounded Borel sets" In this case$X$... 0answers 49 views ### Strongly additive open cover Call an open cover$\mathscr{U}$of a metric space$M$strongly additive if whenever$U,V\in\mathscr{U}$and$U\cap V\ne\emptyset$, then$U\cup V\in\mathscr{U}$. Prove that$M$is compact and ... 0answers 73 views ### A question on pseudocompact space The exercise is from the handbook of set-theoretic topology page 161: Assume$\mathfrak b=\mathfrak c$. Construct a first countable separable zero-dimensional locally compact pseudocompact space ... 0answers 51 views ### Another question in relation to Tychonoff theorem Let$X_i$be compact topological spaces and let$X = \prod_{i \in I}X_i$and let$\mathscr F$be ultrafilter on$X$. Define$\mathscr F_i = \{Y \subseteq X_i : \pi_i^{-1}Y \in \mathscr F\}$. Here$\...
Firstly, I'll give the definitions of sequential compactness and countable compactness. Sequential compactness: If $X$ is a Hausdorff space and every sequence of points of $X$ has a convergent ...