Questions tagged [compactification]

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

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Perfect map and one- point compactification

Let X and Y be non-compact locally compact $ T_{2} $-spaces and f : X → Y continuous map. Define $ f^{*} $ : σ(X) → σ(Y) be a map with : • $ \ f^{*} |{x} $ = f • $ f^{*} $($ \infty_{X} $) = $ \...
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End space of non-compact 2-manifolds in terms of proper rays

I am wondering if the classification of noncompact surfaces given by Ian Richards can be stated with proper rays instead of nested sequences of connected open subsets with compact boundary. Below I ...
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1answer
78 views

Is Stone-Čech compactification the only one with universal property?

The Stone-Čech compactification (also called the Čech-Stone compactification) is the "biggest" compactification of a topological space. (I am working with Hausdorff compactifications only, ...
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1answer
44 views

Uniqueness of one-point compactification, problem with proof [duplicate]

Well known result in general topology is that locally compact Hausdorff space $X$ can be embedded in compact Hausdorff space $Y$ such that $Y = X\cup\{\infty\}$. And this is unique up to homomorphism. ...
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2answers
175 views

What does it mean that "compactification is defined only with respect to the topology of the base space"?

I am reading about topological compactifications, one of the materials I came accross is this paper by Benjamin Vejnar. My question: What does it mean that "the compactification is defined only ...
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1answer
72 views

Show compactification of positive real plane is homeomorphic to $\Bbb R \Bbb P^2$

$\Bbb R \Bbb P^2$ can be thought of as a compactification of $\Bbb R^2,$ and is formed by taking the quotient of $\Bbb R^{3}-\{0\}$ under the equivalence relation $x\sim \lambda x$ for all real ...
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Representing split-complex numbers as intervals and related compactification

Since there is an isomorphism between split-complex numbers and $\mathbb{R}^2$ with element-wise operations, of the following form $a + bj \leftrightarrow (a - b, a + b)$, one can think about a split-...
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1answer
59 views

Stone–Čech compactification of a Tychonoff space can be taken as closure of diagonal mapping image in Tychonoff cube [closed]

Let $X$ - Tychonoff topological space. Show that Stone–Čech compactification of $X$ can be obtained by taking the closure of the image of the space $X$ under the mapping $\Delta_{f\in C(X, I)}f$ in ...
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1answer
13 views

Every locally Compact Hausdorff Space is Regular.

Every locally Compact Hausdorff Space is Regular $Proof$:Let $X=$locally compact+$T_2\implies X^*$ is compact+$T_2\implies X^*$ is Normal($T_4$)$\implies X^*$ is Regular$\implies X$ is regular($T_3$)...
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33 views

Quotient Map From Compactification to One-Point Compactification

Let $X$ be a Hausdorff and locally compact space, and let $Y = X \cup \{\infty\}$ denote its one-point compactification. Let $Z$ be any Hausdorff compactification of $X$. I want to show the following: ...
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1answer
37 views

About butterfly points and non normality.

I'm reading this article and others about the same topic and I found that all autors assume the next theorem. First, a definition. Here $Y^{*}:=\beta{Y}\setminus Y$, i.e., $Y^{*}$ is the remainder of ...
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21 views

Two first countable $T_{3\frac{1}{2}}$ spaces having homeomorphic Stone-Čech compactifications are homeomorphic

It suffices to show that we can reconstruct $X$ from $\beta X$. Every $x \in X$ has $\chi(x, X)\le\aleph_0$. Since $\beta X$ is normal and has $X$ as a dense subspace, the character of every closed ...
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1answer
56 views

Showing that $C_0 (X)^{+} \cong C_0 (X^{+}).$

Let $X$ be a locally compact, non-compact Hausdorff space. Let $X^+$ denote the one point compactification of $X.$ Let $C_0 (X)$ denote the space of all complex valued continuous functions on $X$ ...
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33 views

What does it mean for a space to "carry a natural integration"?

In a ncatlab article, I came across two sentences (on the image below) that are not so clear to me. What does it mean that the compactifications "carry natural integration"? I guess it has ...
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40 views

Prove that the one point compactification of $(0,1)$ is a continuous bijective map to $\mathbb S^1$

Let $X$ be the open unit interval with the usual topology and let $(Y,U)$ be the one-point compactification of $X$, where $Y= X\cup \{p\}$. Prove that the function $f:Y\to S^1$ defined by $f(p)=(1,0)$ ...
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1answer
46 views

A question in compactifications in topology [closed]

I have been taking a course on topology this semester and I need help in this particular exercise. Prove that the function $f: (0,1)\to \mathbb{R}$ defined by $f(x)= \sin(1/x)$ cannot be extended to ...
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34 views

How to construct a homeomorphism with certain conditions on $\mathbb{R}$?

I am reading the paper "Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$ " by Eric Van Douwen. One part of his proof of the Proposition 4 (below) is not clear to me. Suppose $H$...
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100 views

How to prove that $\beta X$ is the only H-compactification of $l^2$?

H-compactification is defined as such compactification that all autohomeomorphisms on the original space can be continuously extended to the compactification. (Sometimes H-compactifications are called ...
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1answer
215 views

Can we determine a number of objects in a category?

Maybe my question is too idealistic, but I try it: Is there any way to look at a category and be able to tell whether this category has zero, one, countable, infinite or any particular number of ...
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31 views

Compactifications of a product of topological space with compact space?

Are there any general properties or rules for a product of a topological space and compact space to admit a compactification? I am usually assuming Hausdorff compactifications, so the original space ...
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67 views

What is the most common name for topological compactifications?

Definition: Topological compactification is such compactification of a topological space such that all autohomeomorphisms on that space can be continuously extended into autohomeomorphisms of the ...
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1answer
27 views

$A_1, A_2 \subset X$ with disjoint closures in $X$, $cl_X(A_1)$ compact $\Rightarrow A_1, A_2$ have disjoint closures in $X_{\infty}$?

Let $X$ be a locally compact Hausdorff space and $A_1,A_2 \subset X$. Suppose that $\overline{A_1}^{X} \cap \overline{A_2}^{X}=\emptyset$ and $\overline{A_1}^{X}$ is compact. Let $X_\infty$ be the one-...
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73 views

Is Stone-Čech compactification an initial object in the category of compactifications?

Let X be a Tychonoff space. Denote $\beta X$ = the Stone-Čech compactification of X. Denote CompHaus = category of compact Hausdorff compactifications of a space. Remark: I consider Hausdorff ...
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31 views

$T=[0, 1]$ with operation $xy=\max \{x, y \}$ has a Folner net?

It is known that every abelian semigroup $T$ does have a Folner net, which means that there is a net $\{F_n\}_{n\in D}$ in $\mathcal{P}_f(T)$, where $\mathcal{P}_f(T)=\{A: A \text{ is finite set of } ...
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2answers
80 views

When a space is NOT locally compact, does it have dense remainder in its compactification?

Any space has to be dense in any of its compactifications (that is part of the definition). Question 1) When the space is NOT locally compact, does that mean that also its remainder in any of its ...
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29 views

Are $\mathbb{R}$, $\mathbb{R^n}$ and $l^2$ open, or closed in their compactifications? What about the remainders?

I hope I formulated the question correctly, if not, I will try to update. Thank you for your help. Are $\mathbb{R}$, $\mathbb{R^n}$ and $l^2$ open, or closed in their compactifications? What about the ...
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131 views

conformal compactification $\overline G$

Construct a conformal compactification, $\overline G$ of $G:=\Bbb R^{1,1}_*$ and/or provide a diagram of the conformal compactification of $G?$ conformal compactification Let $G$ have the metric ...
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1answer
32 views

Proving Tychonoff's theorem using the Stone-Cech compactification

Let $X_i, \ i \in I$ be a collection of compact spaces. Consider the product space $X= \prod_i X_i$. Let $βX$ be the Stone-Cech compactification of $X$. For $i \in I$, let $π_i \colon X \to X_i$ be ...
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2answers
67 views

What is a z-filter?

I am trying to understand meaning of z-filters (def. below). Why are they introduced? Where are they used in mathematics? The only example I have found is that they are used for the construction of ...
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2answers
144 views

How do you visualize Stone-Čech compactification (construction using the unit interval)?

I think I understand the mechanism of constructing Stone-Čech well from Wikipedia. However, I fail when trying to connect this with any concrete examples. For example, for the simplest examples, like $...
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1answer
107 views

Do Stone-Čech compactifications have property that disjoint closed subsets of $X$ have disjoint closures in $\beta X$?

I came across this in Van Douwen´s paper, Characterizations of $\beta \mathbb{Q}$ and $\beta \mathbb{R}$, Proposition 4. Van Douwen writes: "We show that $\gamma H$ = $\beta H$ by showing that ...
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1answer
63 views

How to check universal property when constructing the Stone-Čech compactification?

I read about various constructions of Stone-Čech compactification on Wikipedia. Regarding the construction using unit interval, I have a question about checking the universal property. "In fact, ...
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45 views

Are only the closed subspaces of a compact space compact?

I have read a Wikipedia article about Stone-Čech compactification and have a question about one particular sentence: "... $[0, 1]^C$ is compact since $[0, 1]$ is. Consequently, the closure of $X$ ...
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1answer
64 views

Is there a $T_{3\frac{1}{2}}$ space such that all continuous real-valued functions are bounded?

Is there a non-compact $T_{3\frac12}$ space $X$ (with at least two points) such that all continuous functions $X\to \mathbf R$ are bounded? This is true for $T_3$ spaces as described in this answer: ...
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1answer
51 views

brief explanation of any compactification of a locally compact Hausdorff space X is a quotient space of its Stone-Chech Compactification

I want a simple explanation(not rigors proof; maybe for the rigors proof references are fine) of any compactification of a locally compact Hausdorff space $X$ is a quotient space of its Stone-Chech ...
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1answer
116 views

Why can any homeomorphism on a topological space can be extended to its Čech-Stone compactification?

Let $X$ be a topological space. More times, I came across the statement that the Čech-Stone compactification (the "most general" compactification of a topological space) clearly has a ...
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25 views

Topological compactifications on $l^2$ spaces

I am currecntly studying such compactifications that the homeomorphisms on the original space can be extended onto the compactification (called $topological$ $compactifications$ or $H-...
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What is the Bohr compactification of a topological group and in particular of the real line? How can you define it?

I have encountered the Bohr compactification in the context of Loop Quantum Cosmology and Polymer Quantum Mechanics and I cannot succeed understanding it in a straightforward way. I just got my Master'...
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1answer
43 views

Identify $\overline{\Bbb C^2}$ with $\Bbb S^4\subset\Bbb R^5$

Let $\overline{\Bbb C^2}$ be the one point compactification of $\Bbb C^2$, that is $\overline{\Bbb C^2}=\Bbb C^2\cup\{\infty_2\}$. It is well known that stereographical projection allows to identify $\...
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1answer
29 views

Riemann surface $\Gamma\backslash U$ for some subset $U$ of $\mathbb{C}$ on which a discrete subgroup $\Gamma$ of $SL(2, \mathbb{R})$ acts

I was told relatively vaguely that for some discrete subgroup $\Gamma$ of $SL(2, \mathbb{R})$ (not necessarily contained in $SL(2,\mathbb{Q})$) and a subset $U$ of $\mathbb{C}$, the quotient $\Gamma\...
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68 views

1-point compactification of long line (long circle)?

The long line is locally compact and Hausdorff so it has a 1-point compactification. What can be said about it? Is it anything new? Searching internet for "long circle" returns no results, ...
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$S^k\mathbb{D}^n\simeq \mathbb{D}^{n+k+1}$

I'd like to prove that the $k$-th suspension of the $n$-disk, i.e $\mathbb{D}^n = \left\lbrace x \in \mathbb{R}^n : \| x \| \leq 1\right\rbrace$ is the $n+k$ disk, i.e $$S^k\mathbb{D}^n\simeq \mathbb{...
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19 views

Besides $ω_{(1)}$ what space has Stone-Cech compactification and one-point compactification the same? [duplicate]

For the ordinal space $ω_{(1)}$, the Stone-Cech compactification and the one-point compactification coincide. What is another such space?
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1answer
41 views

Help with proof with One-Point Compactification and Quotient Spaces.

I have been tasked with proving the following: Let $X$ be a compact, Hausdorff space, and let $U$ be a proper opens subset of $X$. Prove that $$ U^{\infty} \cong X / \left( X - U \right)$$ Note that ...
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34 views

Stone–Čech compactification of reals in layman terms

Please, explain me the Stone–Čech compactification of reals without using topological language. I want to understand the concept.
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1answer
43 views

a sequence defined by continuous function on $\mathbb{N} \cup\{ \infty\}$

Let $f$ be a continuous complex-valued function on $\mathbb{N} \cup\{ \infty\}$, one-point compactification of $\mathbb{N}$. Let $\{a_n \}$ be a sequence defined by $a_n =f(n)$ for each $n \in \mathbb{...
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40 views

Compactification of a noncompact regular level set of a map $\Bbb R^n\to \Bbb R$

Consider a smooth function $f:\Bbb R^n\to \Bbb R$, and a noncompact regular level subset $S=f^{-1}(c)$ for some regular value $c\in \Bbb R$. Then $S$ is a smooth hypersurface in $\Bbb R^n$, so it is ...
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1answer
43 views

One- point compactification

Let $X$ be a locally compact, non-compact Hausdorff space. Consider $Y=X\cup \{\infty\}$ as one-point compactification of $X$. Also, we assume that $K$ is compact in $X$ and $U$ an open set of $\infty$...
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1answer
52 views

Show that $X \setminus A$ is locally compact.

Let $X$ be a compact Hausdorff space and $A \subseteq X$ be a closed set. Then $X \setminus A$ is a locally compact Hausdorff space. That $X \setminus A$ is Hausdorff is quite clear as it is a ...
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1answer
117 views

One-Point-Compactification

It is well known that every topological space has a compactification. We therefore consider the one-point extension $(Y, T)$ of a topological space $(X, V)$ which is a compact space with a inclusion ...

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