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Questions tagged [compactification]

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

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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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1answer
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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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1answer
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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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1answer
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If $E$ is a locally compact metric space and $f:[0,∞)→E$ is càdlàg, is $f$ càdlàg as a function into the one-point compactification of $E$ too?

Let $(E,\tau)$ be a locally compact Hausdorff space, $\infty\not\in E$ be an abstract point, $E^\ast:=E\uplus\left\{\infty\right\}$ and $\tau^\ast:=\tau\cup\left\{E\setminus K\cup\left\{\infty\right\}:...
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Relation between Alexandroff extension and density

I have learned Alexandroff extension. Let $X$ be a topological space, and $X^*$ be an Alexandroff extension of $X$. I know that $X$ is dense in $X^*$$\Leftrightarrow$ $X$ is not compact. Yes, it ...
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2answers
39 views

$X$ in its one point compactification

Suppose $X$ is a non-compact space, denote its one point compactification by $X^*$. Since $X$ is open in $X$, thus $X$ is open in $X^*$, can I say $X$ is the interior of $X^*$? Moreover since $X$ is ...
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1answer
53 views

Homeomorphism from punctured sphere to horn torus

Im working on a problem about 1 point compactification, and i am at a step where I want to take the punctured sphere $$ S^{2}\setminus\left\{ (0,0,1)\right\}= \left \{ (x,y,z)\in\mathbb{R}^3 \, \mid \,...
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2answers
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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 ...
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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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2answers
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Continuous extension of a function to the Stone-Cech compactification

Let $C(X)$ be the set of all continuous, real-valued functions on a Tychonoff (completely regular) topological space $X$ and let $\beta X$ be the Stone-Cech compactification of $X$. Now let $f\in C(...
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1answer
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Can a noncompact metric space have a maximal metrizable Hausdorff compactification?

We know the Stone-Čech compactification $(h, \beta X)$ of a Tychonoff space $X$ is its largest (in particular, a maximal) Hausdorff compactification, in the sense that if $(k,\gamma X$) is any other ...
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1answer
34 views

A compact space is its compactification

Referring to this definition of compactification: A pair $(f,Y)$ where $f:X\longrightarrow Y$ is an embedding, $Y$ is compact and $\overline{f(X)}=Y$, is called a compactification of a topological ...
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1answer
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One point compactification of is compact

I am starting to study the one point compactification of a $T_1$ space $S$, namely $S^*=S \cup\{\infty\}$. I understand most of it except the part about $S^*$ being compact. Given $C=\{A/A$ is open ...
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1answer
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Is the algebra map of the ultrafilter monad continuous?

Let $\beta$ be the ultrafilter functor from Sets to Sets, which sends a set $X$ to the set of all ultrafilters on the powerset of $\mathcal{P}(X)$ equipped with its Boolean algebra structure. Then $\...
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1answer
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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 ...
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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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1answer
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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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2answers
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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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1answer
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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 ...
3
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1answer
154 views

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 ...
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1answer
72 views

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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2answers
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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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3answers
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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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0answers
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One-point compactification of disjoint union is homeomorphic to wedge sum of one-point compactifications

I'm trying to prove the following: Let $x$ and $Y$ be locally compact Hausdorff spaces and $X^{*}$ and $Y^{*}$ be their one-point compactifications. Prove there is a homeomorphism $$(X \amalg Y)^{*} \...
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Rational cohomology of section spaces of one point compactification of tangent bundles over closed manifolds.

Let M be a connected closed manifold ( oriented or nonoriented) of finite type (Betti numbers are finite) and $\Gamma(\mbox{TM}_{c})$ is the space of sections of fiberwise one point compactification ...
4
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1answer
183 views

Distinguishing Two Compactifications of $[0,1)$

Pictured below are two subsets of the plane, each a compactification of the closed half-line with remainder a closed arc. I am really frustrated by my inability to prove that the space pictured on ...
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2answers
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If $X$ admits a Hausdorff compactification, then $X$ is locally compact?

I read that if $X$ admits a Hausdorff compactification, that is a compact Hausdorff $C$ such that $X$ is homeomorphic to an open dense subset of $C$, then $X$ is locally compact. Why is that?
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1answer
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Why does $\lim_{n\to \infty} f(n) = 0$ imply that $\beta f=0$ on $\beta \mathbb{N}\setminus\mathbb{N}$

This is a question about $\beta \mathbb{N}$, the Čech-Stone compactification of the natural numbers. I'm reading a proof of the fact that every zeroset in $\beta X \setminus X$ contains a copy of $\...
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1answer
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Čech-Stone compactification of extremally disconnected space

For reference: A space is extremally disconnected iff the closure of every open set is clopen. I am trying to understand a simple proof for the fact that the Čech-Stone compactification $\beta X$ ...
3
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1answer
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Calculation to show $|\mathrm{d}r|^2_{\bar g} = 1$ implies sectional curvatures tend to $-1$.

$\textbf{tl;dr:}$ Given that $r$ is a definining function for the boundary of a conformally compact manifold, how does one show that the sectional curvatures tend to $-1$ if $|\mathrm{d}r|^2_{\bar g} =...
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2answers
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Path connectedness of one point compactification

Let $X$ be a locally compact, metrizable, path-connected space which is not compact. Is the one-point compactification of $X$ also path-connected?
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1answer
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Alexandroff compactification: continuous function extension

Let $(X, \mathcal{T})$ be a non compact topological space, $\infty \notin X$ and $(X^* := X \cup \{\infty\}, \mathcal{T}^* := \{U \subseteq X^*\mid U \cap X \in \mathcal{T} \land (\infty \in U \...
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1answer
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Alexandroff compactification topology

Let $(X, \mathcal{T})$ be a non compact topological space, $\infty \notin X$. We define $$X^* := X \cup \{\infty\}$$ and $$\mathcal{T}^* := \{U \subseteq X^*\mid U \cap X \in \mathcal{T} \land (\...
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1answer
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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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1answer
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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 ...
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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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Is $[0,1]$ the Stone-Čech compactification of $(0,1)$?

In this note by G. Eric Moorhouse, which appears to be some course notes handout, it is stated on page 3: The [two-point] above is the Stone-Čech compactification of $(0,1)≃\mathbb{R}$;that is, $\...
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Show that sequence is eventually constant in discrete space $X$.

Here is the question: Let $X$ be a set with the discrete topology and $\beta X$ be its Stone-Čech compactification. Let $\{{x_n}\}$ be a sequence in $X$ and suppose it converges in $\beta X$, show ...
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1answer
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What are the open sets in Stone-Cech Compactification of $X$?

Let $X$ be a completely regular space (that is, any closed subset and point not in the closed set can be separated by continuous function). Let $C = C(X,[0,1])$ be the collection of continuous ...
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Samuel compactification of the real line

Is there any example of metric t over the real line R, compatible with the Euclidean topology, satisfying that (R, t) is a complete metric space but its Samuel compactification is not homeomorphic to ...
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1answer
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Is there a way to describe these compactifications algebraically?

There is a well known correspondence between locally compact Hausdorff spaces and commutative C* algebras, in which the homoemorphism class of spaces is mapped one-to-one to the isomorphism class of ...
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1answer
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Proof of lemma 2 of “some questions in the theory of bicompactifications”

I am having trouble proving the first claim made in the proof of the following lemma: Lemma 2: Let $X$ be a completely regular space with the bicompactification $Y$. Let $V'$ and $V''$ be two open ...
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1answer
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Does every locally compact subset of $\mathbb {R}^n$ have a one-point compactification on $\mathbb{S}^{n}$?

In my general topology course, we've briefly touched the subject of one-point compactifications (Alexandroff); we've stated that every topological space $X$ has a one-point compactification $(X^+,\...
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Proof of lemma 1 on “some questions in the theory of bicompactifications” by E. Sklyarenko

Lemma 1: Let $\delta$ be a proximity relation on the space $X$ corresponding to the bicompactification $Y$. The bicompactification $Y$ of the space $X$ is perfect with respect to the open set $U$ of $...
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1answer
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Topology generated by $\mathbb{R}$-valued functions of vanishing variation

Let $(X,d)$ be a metric space. Then one knows that the initial topology on $X$ generated by the space of bounded continuous functions $C_b(X)$ with values in $\mathbb{R}$ coincides with the metric ...
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1answer
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Proving the one-point compactification of a topological space is a topology

The following is taken from Introduction to Topological Manifolds by John Lee I am trying to verify that $\mathcal{T}$ is in fact a topology on $X^*$. Note that definition I'm working with for the ...