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Questions tagged [alexandroff-compactification]

For questions concerning the Alexandroff compactification of a locally compact topological space, which is a compact topological space which contains the original space and one extra point.

The Alexandroff compactification of a separated locally compact (but not compact) topological space X is a topological space K which consists of the union of X with a set consisting of a single element (usually denoted by ∞). The open sets of K are the open subsets of X, together with those subsets of K to which ∞ belongs and are such that their complement is compact. This topological space is compact.

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When is Stone-Čech compactification the same as one-point compactification?

For the space $\omega_1$ (with the order topology) we have $\beta\omega_1=\omega_1+1$ (or $\beta[0,\omega_1)=[0,\omega_1]$, if you prefer this notation), i.e., it is an example of a space for which Stone-Čech compactification and one-point…
Martin Sleziak
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Showing one point compactification is unique up to homeomorphism

First for clarity I'll define things as I'm familiar with them: A compactification of a non-compact topological space $X$ is a compact topological space $Y$ such that $X$ can be densely embedded in $Y$ . In particular a compactification is said to…
Serpahimz
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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 call a compactification $\gamma X$ of a space $X$…
Tereza Tizkova
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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 Hausdorff $Y$ is compact $\iota_Y$ is a homeomorphism…
Smiley1000
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Alexandroff extension, projective space and the set of non-zero polynomials of degree at most two

Let be the set of non-zero polynomials $P(x)$ of degree at most two with constant coefficients in $\mathbb{R}$ and a single variable $x$ and possessing real zeros. Given a polynomial $P(x)\equiv{}a\,x^2+b\,x+c$, then the real zeros of this…
Roy
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Characterization of tempered distributions on the sphere $S^n$

Let $\mathbb{R}^n$ be $n$-dimensional Euclidean space and $S^n$ be the $n$-sphere. If we fix a point $x \in S$, the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ can be characterized as the subspace of $C^\infty(S^n)$ whose elements have vanishing…
Keith
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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 two ideas: i) $X$ Hausdorff locally compact, then…
Christian Coronel
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One point compactifition

I have one little problem with one point compactification. Here is example: Let $Z = \alpha \mathbb N = \mathbb N \cup \{\alpha\}$ one point compactification discrete spaces $\mathbb N$ of natural number. I know that open sets are shapes $\{\alpha\}…
Stepan Milosevic
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Is my construction of one-point compactification of $\mathbb{R}$ correct? (+ Clarifying questions)

Could you please check if this construction makes sense and answer to questions to the parts in bold? My construction: The construction can be given explicitly as an inverse stereographic projection. Consider the map $s: \mathbb{R} \rightarrow S^1$…
Tereza Tizkova
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Showing One-point compactification topology on $\mathbb R^n$ is Compact without using Stereographic Projection map.

The following is a well known result from point-set topology: Theorem: Suppose $\mathbb R^n$ is the Euclidean-$n$-space and $\infty$ be a symbol not contained in $\mathbb R^n$(By Russel's Paradox).Consider the set $X=\mathbb R^n\cup \{\infty\}$ by…
Kishalay Sarkar
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Spherical measure $\sigma^n$ on $S^n$ and Lebesgue measure $\lambda^n$ on whole $\mathbb{R}^n$ via stereographic projection

Let $\sigma^n$ be the spherical measure on $S^n$ and $\lambda^n$ be the Lebesgue measure on $\mathbb{R}^n$. Let us fix a point $a \in S^n$ and consider the stereographic projection $\Phi : S^n - \{a \} \to \mathbb{R}^n$. By slight abuse of notation,…
Keith
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Alexandroff compactification for closed right half plane

As title says, I want to find a subspace of $\mathbb{R^2}$ or $\mathbb{R^3}$ homeomorphic to the Alexandroff compactification of $$X = \{(x,y) \in \mathbb{R^2} \colon x \geq 0\}$$ I have been trying to find a homeomorphism between $X$ and a subspace…
David
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Compactification of a topological semigroup with right-identity structure

I'm looking for a hint on this question: Let $T$ be as a semigroup with right-identity structure i.e. $rs=r$ for all $r, s\in T$. Also, we consider $T$ as a topological semigroup that is a locally compact, noncompact Hausdorff space. Define…
user479859
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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 an open subset $U$ of $X$ such that…
shoteyes
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How might I resolve a failure in my own attempt at showing that Alexandroff's extension is compact? Finite intersection property seems to fail

$\newcommand{\F}{\mathcal{F}}\newcommand{\O}{\mathcal{O}}\newcommand{\id}{\operatorname{Id}}$Let $X$ be a nonempty topological space. Let $\omega$ be an arbitrary object, taken to be not in $X$, and let $X^\ast=X\sqcup\{\omega\}$. The topology on…
FShrike
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