Questions tagged [compactification]
Use this tag for questions about making a topological space into a compact space.
294
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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 ...
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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 $∞$...
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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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If the ultrafilter space is Hausdorff, must the base space be discrete?
The question is in the title. Most of this post is a contextual preamble and some of my thoughts on the matter, I have not made much solid progress (and don't even know if this is true...)
$\...
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Stone Cech compactification of $(0,1]$is not $[0,1]$
If I considered $(0,1]\subset \beta(0,1]=[0,1]$ this is trivial because the universal property would mean we would need a continuos function on $[0,1]$ with $g(t)=\sin(1/t)$ fot all $t \in(0,1]$ which ...
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Continuity of map $SU(2)\to \mathbb{C}\cup \{\infty\}$
Let $Y:=\mathbb{C}\cup\{\infty\}$ be the one-point compactification of $\mathbb{C}$. Define
Consider the map
$$\Phi: SU(2)\to Y: \begin{pmatrix}a & b\\ c & d\end{pmatrix}\mapsto \frac{b}{a}$$
...
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Prove that the Stone-Cech compactification of a discrete space is such that the closure of every open set is open [duplicate]
Let $X$ be a discrete space and $\beta (X)$ its Stone-Cech compactification. We have to prove that for every open set $U$ of $\beta (X)$, $\overline{U}$ is open in $\beta (X)$.
So far the only idea I ...
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$l_\infty $ and Stone-Cech compactification of $\mathbb{N}$
$l_\infty$ is the vector space of real bounded sequences with the norm $$d(x,y)=\sup\{|x_n-y_n|, n\in \mathbb{N}\}.$$ I need to show that there is an isomorphism $T$ between $C(\beta \mathbb{N})$ and $...
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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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One-point compactification is compact
Let $X$ be a locally compact Hausdorff space and $Y = X \cup \{\infty\}$ its one point compactification. The following is Munkres' proof that $Y$ is compact.
Let $\mathscr{A}$ be an open covering of $...
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Convergence of sequences in the one-point compactification
Consider a 1st countable and Hausdorff space $X$ and its one-point compactification $X^* = X \cup \{\infty\}$. The topology $\mathcal{T}^\ast$ contains the topology $\mathcal{T}$ of $X$ and all sets ...
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How to understand the compactification of the upper half plane as motivation for Gromov compactification of an arbitrary $\delta$-hyperbolic space??
Let $X$ be a $\delta$-hyperbolic space. Let us define an equivalence relation on geodesic rays $\gamma_1, \gamma_2 :[0,\infty) \rightarrow X$ by {$\gamma_1 \sim \gamma_2\,\, \iff d(\gamma_1(t),\...
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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 ...
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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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understanding inverse image in topology
the exercise I was reading was finding one-point compactification of $(0, 1)$. the solution convinced me that was $S^1$
The problem is ths: It seems intuitive that $f^{-1}(V)$ is the set $Y-[\...
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Question on the proof of the theorem 29.2 in Munkres' topology textbook
The Theorem 29.2 in Munkres' Topology textbook states as follows:
Theorem 29.2. Let $X$ be a Hausdorff space. Then $X$ is locally compact if and only if given $x \in X$, and given a neighborhood $U$ ...
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Continuous open image of locally compact space
Let $(X,\tau)$ be a topological space. A subspace is called precompact if its closure is compact. Now, local compactness
Definition 1 (Munkres): Every point $x$ has an open neighbourhood $U_x$ ...
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Equivalent criterion of local compactness
I'm trying to understand the proof of the following criterion:
Let $X$ be a Hausdorff's space and locally compact in $x\in X$, then for all $U$ open that contains $x$ exists an open set $V$ such that $...
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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 ...
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What is the cardinality of the inverse image of a point under the map $β(ℕ^2)→β(ℕ)^2$?
I will write $β$ for the Stone-Čech compactification. Let $p$ be the canonical map $β(ℕ^2)→β(ℕ)^2$. Let $(u,v) ∈ β(ℕ)^2$. If $u$ or $v$ is in $ℕ$, then $p^{-1}(u,v)$ is a singleton. Otherwise, $p^{-1}(...
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Compactification of compact space
Let $(Y,h)$ be a compactification of a compact Hausdorff space $X$, prove that $h(X)=Y$.
My attempt. Since $(Y,h)$ is a compactification of $X$, we have that $Y$ is a compact space and the function $g:...
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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 ...
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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 ...
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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 ...
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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: ...
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Closure of Y in one-point compactification
I have some trouble with following problem:
Let X be a locally compact Hausdorff space where $X^+=X \cup \{\infty\}$ is its one-point compcatification and $Y \subseteq X$. Compute the closure of Y in $...
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One point compactification and closed subspace
Im having some trouble on where to start with the following problem:
Let X be a locally compact Hausdorff space where $X^+=X \cup \{\infty\}$ is its one-point compcatification and $Y \subseteq X$. ...
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Compactification of the upper half plane
Consider the upper half plane $\mathbb{H} = \{ z \in \mathbb{C} : \text{im}(z) > 0 \}$ as a Riemann surface.
I suspect that the compactification of $\mathbb{H}$ is the Riemann sphere $\mathbb{P}_{\...
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