Questions tagged [compactification]
Use this tag for questions about making a topological space into a compact space.
319 questions
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Prove that the inclusion of the open disk into the closed disk is a compactification and that it is not the Alexandroff compactification. [closed]
Could someone help me demonstrate the following?:
“Prove that the inclusion of the open disk into the closed disk is a compactification and that it is not the Alexandroff compactification.”
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On commutative $C^*$-algebras and compactifications
A C$^*$-algebra $\mathcal{A}$ is a $\mathbb{C}$-algebra together with a norm $\|\cdot\|$ with respect to which $\mathcal{A}$ is complete and the norm is submultiplicative ($\|xy\|\leq \|x\|\|y\|$). ...
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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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Question about one-point compactification
Let $T = (S,\tau)$ be a non-empty topological space.
Let $p$ be a new element not in $S$.
Let $S^∗:=S\cup\{p\}$.
Let $T^∗=(S^∗,\tau^∗)$ be the Alexandroff extension on $S$.
I want to show that $T^*...
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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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Open Problem on $Aut(\mathcal{P}(\omega)/fin$
In Jan van mills "Problems on $\beta \mathbb{N}$" in question 4 they ask:
"Can one have non-trivial autohomeomorphisms but only very
mild ones; the set of points where an ...
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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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Extending isotopy to the one-point compactificaton of $\mathbb{R}^n$
Exercise 12.6 in Adams and Franzosa's "Introduction to Topology: Pure and Applied" suggests proving that if two knots are equivalent in $\mathbb{R}^3$, then they are equivalent in $S^3$ (...
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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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Homogeneity of Stone-Cech compactification
A topological space $X$ is said to be $\textit{homogeneous}$ if to every pair of points $p$ and $q$ of $X$, there exists at least one homeomorphism of $X$ which carries $p$ to $q$.
Suppose $\beta X$ ...
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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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Hawaiian Earring vs. My Space: Fundamental groups
While thinking about the Hawaiian earring space $H$, I thought of a two point compactification of the union of a family of disjoint open intervals $$\large\large\amalg_{n=1}^\infty(0,1)$$ call this $J$...
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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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The category of compactifications
Fix a Hausdorff space $X$. Let $\mathcal{C}_X$ be the category of compactifications of $X$:
The objects of $\mathcal{C}_X$ are spaces $Y$ with a mapping $\iota_Y: X \to Y$ such that:
$Y$ is ...
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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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Can space be augmented with a plane at infinity so that parallel planes intersect at a line at infinity?
The real plane can be augmented with a line at infinity such that two parallel lines intersect at a point at infinity, and the set of all such points forms a line.
In space (3 dimensional solid ...
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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 $\mathbb{N} \times \mathbb{N}$
Does there exist a compactification $X$ of $\mathbb{N} \times \mathbb{N}$ with the following properties?
$X$ is compact
$X$ is Hausdorff
$\mathbb{N} \times \mathbb{N}$ is dense in $X$
$X \setminus ...
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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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3
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One point (non-Hausdorff) compactification of compact space
A compactification of a space $X$ is an embedding $f:X \to Y$ so that (1) $Y$ is compact, (2) $f(X)$ is dense in $Y$. If furthermore, $Y\setminus f(X)$ is a single point, we say it is a one point ...
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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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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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$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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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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2
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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:...
