# Questions tagged [compactification]

Use this tag for questions about making a topological space into a compact space.

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### One Point Compactification and Stone-Čech compactification question

'Let $X$ Hausdorff locally compact space such that every continuous map $f: X \to \mathbb R$ can be extended to a continuous map $g: X^{\ast} \to \mathbb R$. Prove that $X^{\ast} = \beta (X)$. I have ...
1 vote
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### Cohomology of Hawaiian earring, Hatcher exercise

Hatcher 3.3.21 (quoted below for completeness): For a space $X$ , let $X^+$ be the one-point compactification. If the added point, denoted $\infty$, has a neighborhood in $X^+$ that is a cone with $∞$...
1 vote
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### Definition of proper map and compactification? [duplicate]

A proper map is a continuous map such that compact sets have compact preimage. For locally compact Hausdorff spaces, it seems that a proper map can be extended to a continuous map between one-point ...
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### Extension of embeddings with Stone–Čech compactification

Given any two topological spaces $X$ and $Y$, and given any continuous function $f:X\to Y$, There exists a unique extension $\beta f:\beta X\to \beta Y$. If it is also given that $f$ is an embedding - ...
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### Alternative Compactification of $\mathbb{C}$

Of course, there is the compactification of $\mathbb{C}$, $\mathbb{C}\cup\{\infty\}$, which allows us to correspond the complex plane to the unit sphere. Is there any use in compactifying $\mathbb{C}$ ...
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### Universal property of one-point-compactification

In this question, we define locally compact to be Hausdorff and every point has a compact neighbourhood. Similarly, we define compact to be Hausdorff and every open cover has a finite subcover. Then ...
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### Is the one-point compactification of the rationals sequential or Frechet-Urysohn?

Let $X=\mathbb Q^*=\mathbb Q\cup\{\infty\}$ be the one-point compactification of the rationals. The open sets in $X$ are the open sets in $\mathbb Q$, together with the complements in $X$ of the (...
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### Weight of the Stone-Cech compactification of a Tychonoff space

Let $w(X)$ be weight of $X$, that is least infinite cardinality of a basis of $X$. Here $X$ is assumed to be Tychonoff. Is the inequality $w(\beta X)\leq 2^{w(X)}$ true? This seems to hold for many ...
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### Fourier / Gelfand transform vanishes at infinity?

I've come across the fact that the Fourier transform (or, more general, the Gelfand transform) vanishes at $\infty$. See for example "Principles of Harmonic Analysis" by Deitmar and ...
1 vote
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### Compact Manifold Realized as One-Point Compactification

I'm working on the following Problem from Lee's Introduction to Topological Manifolds: "Let $M$ be a compact manifold of positive dimension, and let $p \in M$. Show that $M$ is homeomorphic to ...
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### Universal property of compactification without separation axiom assumption

Let $X$ be a topological space let $Y$ be a compact space (we don’t know if $Y$ satisfies any of the separation axioms) with an compactification $\varphi \colon X \to Y$. Assume that for every compact ...
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### Stone-Čech Compactification of disjoint union as adjoint functor

Is it true that the Stone-Čech compactification preserves disjoint union (even infinite disjoint unions)? That is, is it true that $\beta(\bigsqcup_\alpha X_\alpha) = \bigsqcup_\alpha \beta X_\alpha$? ...
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### Stone–Čech compactification vs Stone Duality

Let $B$ be a Boolean algebra. There are two ways to turn $B$ into a topological space. View $B$ as a discrete space and take the Stone-Cech compactification resulting in $\beta(X)$. This is the same ...
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### $(\bigcap_{\lambda\in\Lambda_1} C_\lambda)\,\bigcap\,(\bigcap_{\lambda\in\Lambda_2} A_\lambda)$ is a compact closed subspace of $S$. Is my proof ok?

I am reading a proof of a proposition about compactification. In the proof, I guess the author uses the following proposition. Proposition 1: Let $S$ be a topological space. Let $C_\lambda$ be ...
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### Maximal Non-Hausdorff Compactification

I have recently started to read about compactifications of topological spaces, however I would like to clear my mind on a few things. For starters, I am interested in generic topological spaces (not ...
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### Non-equivalent compactifications of $[0, \infty)$

I need to find non-equivalent compactifications for the interval $[0,\infty)$. I found the basic compactification which is analogy for inverse of stereographic projection only for one dimension and ...
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### Why does every infinite closed subset of $βℕ$ contains a copy of $βℕ$?

In this answer of Andreas Blass to a question of mine, it is said that every infinite closed subset of $βℕ$ contains a copy of $βℕ$. I have a proof, but I think it can be improved and would like to ...
1 vote
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### One-point compactification of $S^3\setminus S^1$ [closed]

Let $S^1$ be a circle embedded in $S^3$. Is the one-point compactification of $S^3\setminus S^1$ homeomorphic to $S^3$?
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### How is it that $X \subset \beta(X)$ and $\overline{X} = \beta(X)$?

My understanding of the Stone-Cech Compactification of a space $X$ is as follows: Let $\mathcal{F} = \{ f:X \to I_f \}$ so that $f$ is continuous and $I_f \subset \mathbb{R}$ is a compact interval. ...
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### If $X$ is disconnected, then $\prod_{f \in \mathcal{F}} I_f$ is disconnected

I'm currently trying to prove a statement about the relationship between the connectedness of $X$ and the connectedness of $\beta(X)$. But the nature of this question regards a specific detail. Let $X$...
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### Is the Alexandroff extension of a locally compact, second-countable space second-countable?

If $X$ is a locally compact, second-countable topological space, then is its Alexandroff extension $X^*$ also second-countable? Our definition of locally compact is that for every $x$ in $X$, we have ...
1 vote
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### Why are two descriptions of strong zero-dimensionality equivalent?

Def: A Tychonoff space $X$ is said to be strongly zero-dimensional if its Stone-Čech compactification $\beta X$ is totally disconnected (that is if the only connected subspaces of $\beta X$ are ...
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### Why does $\mathbb{N}$ have only two topological compactifications?

Definition: A compactiﬁcation $\gamma X$ of a space $X$ is said to be a topological compactiﬁcation if all autohomeomorphisms of the space $X$ can be continuously extended to a mapping of $\gamma X$ ...
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### Cancellation property of the Stone Cech compactification

Let $G$ be a discrete countable group and let $\beta G$ be the Stone-Cech compactification of $G$, which has the structure of a semigroup. Is $\beta G$ left cancellable? What about right cancellable? (...
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### Does a Compact Subset Remain Compact in the Alexandroff Compactification?

Let $X$ be a topological space that enjoys the Hausdorff property, $K \subset X$ be compact, and $\widetilde{X}$ be the Alexandroff compactification of $X$. Is it true that $K \subset \widetilde{X}$ ...
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### Proof that $\alpha \mathbb{R}$, the two-point compactification and $\beta \mathbb{R}$ are the only three topological compactifications of $\mathbb{R}$

My question is: Is my proof of the proposition from Van Douwen´s paper called Characterization of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$ correct? Claim: $\alpha \mathbb{R}$, the two-point ...
1 vote
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### Understanding a detail of a proof about topological compactifications

I don't completely the following proof from Van Douwen's paper. My exact questions are below the proof. Van Douwen proves that If $X$ is a non-compact strongly zero-dimensional space in which every ...
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### How to prove that the Stone-Čech compactification of a Tychonoff space always exists?

I am wondering how to prove the following theorem: Let "X" be a Tychonoff space. Then its Stone-Čech compactification exists and it is unique (up to homeomorphism). The uniqueness part is ...
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### Stone–Čech compactification of a space removing a point in a compact space

Let $X$ be a compact Hausdorff space and $x \in X$ be a non-isolated point. In the proof of Lemma 3.1 in "Norming $C(\Omega)$ and Related Algebras" by B. E. Johnson it seems to be asserted ...
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### continuity of a map into a one-point compactification

Let $(E,\tau)$ be a topological space, $\Delta\not\in E$, $E^\ast:=E\cup\{\Delta\}$ and $$\tau^\ast:=\tau\cup\{E^\ast\setminus B:B\subseteq E\text{ is }\tau\text{-closed and }\tau\text{-compact}\}.$$ ...
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### Question about the proof about topological compactifications of $\mathbb{R}$

I have a question regarding the proof about compactifications of $\mathbb{R}$. I am reading Van Douwen´s paper Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$, where he defines: We ...
1 vote
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### If the Stone-Čech compactification $\beta X$ of $X$ is connected then $X$ is connected

So I am searching to prove that if the Stone-Čech compactification $\beta X$ of a Tychonoff space $X$ is connected then $X$ is connected. Therefore, I tried to prove that $\beta[X]$ is connected and ...
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### Can we determine an explicit construction of the functions $\beta(f)$, where $\beta$ is the Stone-Cech adjoint functor?
$\newcommand{\top}{\mathsf{Top}}\newcommand{\ch}{\mathsf{CptHaus}}\require{AMScd}$This is a question about the Stone-Cech compactification, from the perspective of category theory. The TLDR is that: ...