Mathematics

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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Showing that the one-point compactification of an open set in the Euclidean plane is path-connected

Problem: Let $U\subseteq \mathbb{R}^2$ be any open set in the Euclidean plane $(\mathbb{R}^2$ with the standard topology) and consider $(U,T)$ as a topological space with the subspace topology inherited from the plane. Prove or disprove that $(U^*,…
Dooley
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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 C haracterizations 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…
Elia Immanuel Auer
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Exercise 4.47, Folland's Real Analysis, Alexandroff compactification

Let $X$ be a locally compact Hausdorff space (LCH). Let $\infty$ denote an element that is not in $X$ and $X^\ast=X\cup\{\infty\}$ . Let $\mathcal{T}$ be the collection of all subsets of $X^\ast$ such that either (i) $U$ is an open subset of $X$ ,…
psie
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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,…
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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Exercise on Alexandroff compactification

Let $X,Y$ be Hausdorff and locally compact topological spaces, with $Y$ not compact, and let $\hat{Y}$ be the Alexandroff compactification of $Y$ . Let $A\subset X$ be an open, relatively compact subset of $X$ and consider a map $f:A\to Y$ . Denote…
Steppenwolf
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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 $X=\mathbb R^n\cup \{\infty\}$…
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…
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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