Questions tagged [compactification]
Use this tag for questions about making a topological space into a compact space.
333 questions
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Showing that Smirnov compactification is a compactification
This is part of the proof that proximities on a topological space $X$ are order-isomorphic to compactifications of $X$. See wikipedia for definition of proximity space. Here I am defining proximity ...
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Uniqueness of Bohr compactification
The Bohr compactification is defined e.g. here.
The following is theorem is easy (hence nobody bother to prove it).
Theorem. The Bohr compactification exists and is unique up to isomorphism.
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Cardinality of $\beta \mathbb{N}$ without using the ultrafilter definition
I'm currently studying compactifications in general, but i've been stuck on how to exactly prove that the cardinality of $\beta \mathbb{N}$ is $2^\mathfrak{c}$. I've seen people using the ultrafilter ...
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Separation in perfect compactifications
I refer to the book Isbell, Uniform spaces, 1970.
p.97. Definition. A set $C$ separates $A$ and $B$ in a space $X$ if $X\setminus C=M\cup N$ where $M,N$ are separated sets containing $A,B$. And $M,N$ ...
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Quantifying the size of Stone-Cech compactification of the real line
We know that the Stone-Cech compactification is huge but I want to know exactly how huge. Let's take $\mathbb{R}$ for simplicity. Then let $C$ be the set of all bounded continuous functions from $\...
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Topological spaces with dense discrete subspaces
Let $X$ be a totally disconnected compact Hausdorff space. Does $X$ always have a discrete dense subspace $Y$?
My motivation for asking this is that the Stone-Čech compactification of a discrete space ...
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Sizes of compact, separable spaces / compactifications of the integers
Update (2025-01-07)
I have just cross-posted this question at MO.
In the following, all topological spaces are Hausdorff.
Let $\kappa$ be a cardinal, $\kappa \le 2^\mathcal c$.
The following are ...
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A simply connected space has a simply connected compactification and non-simply connected compactification?
$\mathbb{R}$ is an example of a simply-connected space with non-simply connected compactification (one-point compactification of $\mathbb{R}$ is $\mathbb{S}^1$ so not simply connected.) but 2-point ...
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Does $C\times \omega_\alpha$ admit a positive dimensional compactification?
Let $C$ be the Cantor set and $\alpha > 0$ a non-limit ordinal with $|C| = \mathfrak{c} < \aleph_\alpha$. Consider the space $C\times \omega_\alpha$.
By the exercise 9K in Gillman and Jerison, $\...
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Equivalent characterizations of Stone-Cech compactification and generalization to Wallman-Frink compactifications
In the book Normal topological spaces by Alo and Shapiro the following theorem is present:
Theorem. If $X$ is a dense subspace of Tychonoff space $Y$ then the following are equivalent:
Every ...
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Autohomeomorphisms of Stone-Cech remainder
I really want to read into autohomoeomorphisms of the stone cech remainder of $\mathbb{N}$. Does anyone have some references/papers that really go into detail on this topic? Preferably one/s that ...
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Growth of Stone-Cech compactification
Let $X$ be a Tychonoff space and $\beta{X}$ be the Stone-Cech compactification of $X$. The set $\beta{X}\setminus X$ is known as the growth of $X$ in $\beta{X}$.
Is there any reference which discusses ...
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Infimum for family of compactifications of a topological space
We saw in our topology course if a topological space admits one Hausdorff compactification (so is $T_{3\frac{1}{2}}$) then any family of compactifications admits a supremum in the natural order on ...
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Stone-Cech compactification via lattice ideals of $Coz(X)$
While studying P. T. Johnstone's book Stone Spaces I have come across the following Proposition (Section: 3.3)
Where $max C(X)$ is the set of all maximal ideals of $C(X)$ (ring of real-valued ...
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When can a homotopy lift to its compactification?
Let $h_t:X\to Y$ be a homotopy, we assume both spaces are locally compact and hausdorff, and each $h_t$ is proper, when can we lift it to a homotopy of one point compactification $\bar X\to \bar Y$? ...
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Stone–Čech Compactification always exists?
i have a question regarding the Stone–Čech compactification of some topological space.
On Wikipedia Page, it says "A form of the axiom of choice is required to prove that every topological space ...
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3-Point Compactification of $\mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0}$
Does there exist a compactification $X$ of $\mathbb{R}_{\geq 0} \times \mathbb{R}_{\geq 0}$ with the following properties?
$X$ is compact
$X$ is Hausdorff
$\mathbb{R}_{\geq 0} \times \mathbb{R}_{\...
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Map from Compactification to Original Space
If one has a map $f$ from a topological space $X$ to another space $Y$ and then one takes the compactification of $Y$ (for example, if $Y = \mathbb{R}^n$ the compactification is constructed by taking ...
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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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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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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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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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$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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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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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),\...