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Compactifications of discrete spaces The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer} with the order topology.A sequence $${\displaystyle \{a_{n}\}}$$ in a topological space $${\displaystyle X}$$ … See more
In the mathematical field of topology, the Alexandroff extension is a way to extend a noncompact topological space by adjoining a single … See more
Let $${\displaystyle (X,\tau )}$$ be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of $${\displaystyle X}$$ obtained by adding a single point, which could also be called … See more
Let $${\displaystyle c:X\hookrightarrow Y}$$ be an embedding from a topological space X to a compact Hausdorff topological space … See more
• Bohr compactification – compact Hausdorff group associated to a topological group
• See moreWikipedia text under CC-BY-SA license Consider the real line with its ordinary topology. This space is not compact; in a sense, points can go off to infinity to the left or to the right. It is possible to turn the real line into a compact space by adding a single "point at infinity" which we will denote by ∞. The resulting compactification is homeomorphic to a circle in the plane (which, as a closed and bounded subset of the Euclidean plane, is compact). Every sequence that ran off to infinity in the real line will then converge to ∞ …
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WEBEvery Alexandrov topology is locally compact in the sense that every point has a local base of compact neighbourhoods, since the smallest neighbourhood of a point is always …
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WEBMay 11, 2024 · This one-point compactification is also known as the Alexandroff compactification after a 1924 paper by Павел Сергеевич Александров (then …
WEBJan 11, 2018 · The unique Hausdorff compactification $\alpha X$ of a locally compact, non-compact, Hausdorff space $X$, obtained by adding a single point $\infty$ to $X$. …
WEBMar 3, 2013 · Given a topological space X, we wish to construct a compact space Y by appending one point: This is called a one-point compactification of X. Examples. A one-point compactification of (0, 1) …
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WEBVolume 107, Number 4, December 1989. METRIZATION OF THE ONE-POINT COMPACTIFICATION. MARK MANDELKERN. (Communicated. by James E. West) …
WEBOct 1, 2012 · Using the Alexandroff one-point compactification as a point of departure, we study a general procedure for building an extension 〈 X ∪ I, τ 0 〉 of a topological …
WEBFeb 10, 2018 · The Alexandrov one-point compactification of a non-compact topological space X is obtained by adjoining a new point ∞ and defining the topology on X ∪ {∞} to …
WEBone-point compactification. The following theorem characterizes the spaces with one-point compactifications. Theorem 2.3 A space has a one-point compactification iff is a …
Polyadic space - Wikipedia
WEBIn mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification …
general topology - Alexandroff one-point compactification
WEBJan 9, 2015 · The one point compactification is not an open ball. I suggest going down one dimension as that allows an easier visualization. The Alexandroff compactification …
general topology - Alexandroff one point compactification
WEBTheorem: if $X$ is compact Hausdorff and $p \in X$ is a non-isolated point of $X$, then $X$ is the one-point compactification of $X\setminus\{p\}$ (or $X - \{p\}$ if you prefer that). …
[PDF] Alexandroff One Point Compactification - Semantic Scholar
WEBIn the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact …
Alexandroff One Point Compactification - EuDML
WEBIn the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. Some properties of the locally compact …
Compact space - Wikipedia
WEBAny locally compact Hausdorff space can be turned into a compact space by adding a single point to it, by means of Alexandroff one-point compactification. The one-point …
Alexandroff One Point Compactification - Semantic Scholar
WEBAlexandroff One Point Compactification In the article, I introduce the notions of the compactification of topological spaces and the Alexandroff one point compactification. …
Long line (topology) - Wikipedia
WEBIn topology, the long line (or Alexandroff line) is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has …
Locally compact space - Wikipedia
WEBLet X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.e., there exists an open set U and a compact set K, …
Metrizability of One-Point (Alexandroff) Compactification
WEBAug 9, 2020 · Let $X$ be locally compact metric space, and $X^*$ be its one-point compactification, with the extra point being $\{\infty\}$. Then, the following are …
general topology - Non-Hausdorff one-point compactifications ...
WEBSep 7, 2020 · A compactification $X^*=X\cup\{\infty\}$ obtained by adding a single point will be called a one-point compactification. ( Wikipedia seems to reserve that term for …