# Questions tagged [compactification]

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

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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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### 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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### 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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### 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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### 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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### 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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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 ...