# Questions tagged [compactification]

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

114 questions
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### A $T_{3 \frac{1}{2}}$ Lindelöf space $X$ is normally placed in $\beta X$

A subset $X \subset Y$ is normally placed in $Y$ if whenever $X \subset U$ for $U$ open in $Y$ there are $F_n$ closed in $Y$ such that $X \subset \bigcup_{n \in \mathbb{N}}F_n \subset U$. Let $X$ ...
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### Example 4, Sec. 29, in Munkres' TOPOLOGY, 2nd ed: How is the one-point compactification of the real line homeomorphic with the circle?

Here is Theorem 29.1 in the book Topology by James R. Munkres, 2nd edition: Let $X$ be a [topological] space. Then $X$ is locally compact Hausdorff if and only if there exists a [topological] space ...
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### Maxap groups with zero-dimensional group compactifications

Suppose $G$ is a countable group. We say that $G$ is maxap if there is an injective homomorphism $\phi: G\to K$ with $K$ a compact group. My question is what we can demand of $K$. Given $G$ a maxap ...
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### Stone-Čech compactification is extremally disconnected.

If $X$ is a discrete topological space, one can realize its Stone-Čech compactification by means of ultrafilters. The compactification can be characterized in terms of its universal property. I want ...
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### Intuition for the Stone-Čech compactification via ultrafilters

Definitions used: Given some set $X$, denote by $\beta X$ the set of ultrafilters on $X$. We can view $X$ as a subset of $\beta X$ by identifying each point $x \in X$ with the principal ultrafilter ...
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### One-point compactification and algebraic geometry

An affine curve $C:\{(x,y)\in\mathbb{C}^2:Q(x,y)=0\}$ can always be extended to a projective curve $\tilde{C}:\{[x:y:z]\in\mathbb{CP}^2:P(x,y,z)\}$ where $P(X,Y,Z)$ is a homogenisation of the ...
1answer
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### For the existence of one-point compactification, do we need locally compactness?

In the book Topology by Munkres, at page 184, it is given the existence and uniqueness of one point compactification of a locally compact Hausdorff space; however, in the existence part, I can't see ...
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### One-point compactification problem 2

I know that the one-point compactification of $X,$ a topological space, is $Y=X\cup\{\infty\}$ and $\text{Top}_Y=\text{Top}_X\cup\{Y\setminus K: K$ compact in $X \}.$ Question, is $Y$ compact? My ...
1answer
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### How does the product construction of the Stone-Cech compactification work?

Wikipedia's article on the Stone-Cech compactification gives several constructions of it, one which is this: One attempt to construct the Stone–Čech compactification of $X$ is to take the closure ...
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### Does every compactification have an extension property?

A pair (Y, f) is a compactification of a topological space X iff for every compact space X' and continous map g from X to X', there is a continuous map g' from Y to X' such that g'(f(x))=g(x) for all ...
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### Conformal compactification of spacetimes

I’m looking answers and/or references concerning the following questions about conformal compactification. Given a $d$-dimensional spacetime $(M,g)$ (or simply just for $d=4$): Does the conformal ...
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### Does there exist a continuous bijection from real line onto the unit circle? [duplicate]

Does there exist a continuous bijection from $\mathbb R$ onto $S^1$ ? I know that there isn't any continuous bijection from $S^1$ onto $\mathbb R$ because such a continuous bijection would be a ...
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### One-point compactification and the existence of a contractible open neighborhood of infinity

My question is related to this one. I would like to know what extra conditions a locally compact (but not compact), Hausdorff space should satisfy such that in its one point compactification the point ...
1answer
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### What is the Stone space of the free sigma-algebra on countably many generators

The Stone space of the free Boolean algebra on countably many generators is the Cantor space $2^\omega$. What is the Stone space of the free (Boolean) $\sigma$-algebra on countably many generators?
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### Translation in one-point compactification

I'm trying to determine when translating a subset of a space is homeomorphic to the original space. This obviously isn't always possible since connectedness is a topological property. The motivation ...
1answer
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### Gelfand-Naimark theorem and compactifications

Let $X$ be a Hausdorff space and let $C(X)$ denote the set of all continuous and bounded functions from $X$ to $\mathbb{C}$. It is a well-known fact that $C(X)$ forms a unital and abelian $C^*$-...
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### Show that $Cl_{\beta\mathbb{N}}\{0,2,4,…\}$ is open in $\beta\mathbb{N}$

Let $\beta\mathbb{N}$ denote the Stone-Cech compactification of the space $\mathbb{N}$ of the natural numbers with the discrete topology and let $E$ denote the set of even natural numbers. Show that ...
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### Is the diagonal of $X \times X$ closed in $\bar{X} \times \bar{X}$?

Let $X$ be a topological space. Let $\bar{X}$ denote a compactification of $X$ (i.e., $\bar{X}$ is a compact Hausdorff space such that $X$ is an open dense subspace of $\bar{X}$). Notice that in ...
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### ultrafilter convergence versus non-standard topology

I have recently been reading about the non-standard characterisation of topological spaces, by saying which points of ${^*X}$ are infinitesimally close to which standard points. The theory looks a lot ...
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### Stone-Cech compactification of locales

I'm reading 'Stone-Cech compactification of locales I' by Banaschewski and Mulvey. I have a question about lemma 6: I don't understand why it suffices to show that the ideals $h(x)$ and $g(x)$ are ...
2answers
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### Why is an infinite compact Hausdorff space with a dense countable subset a compactification of $Z^+$?

Studying General Topology from Munkres, I just read about Stone-Čech compactifications for the first time, tried to solve the exercises provided by the author and I came across the following one: "...
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