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Compact space - Wikipedia
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all limiting values of points. For example, the open interval (0,1) … See more
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) … See more
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite See more
• A compact subset of a Hausdorff space X is closed.
• In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff … See moreAny finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite number of distinct … See more
• Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
• Any space carrying the cofinite topology is compact. See moreWikipedia text under CC-BY-SA license Locally compact space - Wikipedia
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood.
In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces.Wikipedia · Text under CC-BY-SA license- Estimated Reading Time: 9 mins
Compactly generated space - Wikipedia
WEBIn the simplest definition, a compactly generated space is a space that is coherent with the family of its compact subspaces, meaning that for every set , is open in if and only if is …
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Locally Compact Spaces - Mathematics and Such
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) …
Compact space - Encyclopedia of Mathematics
WEBJun 5, 2020 · Compact space. A topological space each open covering of which contains a finite subcovering. The following statements are equivalent: 1) $ X $ is a non-empty …
A question about local compactness and $\\sigma$-compactness
WEBDec 31, 2016 · The desired property is called the existence of an exhaustion by compact sets for the space X X. (see Wikipedia) The answer of @bof holds in the context of …
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Compact and Locally Compact Spaces | SpringerLink
WEBNov 17, 2022 · The concept of compactness of a set is pervasive in analysis. This chapter is devoted to the study of compact spaces and locally compact spaces. The concept of …
How to understand the concept of compact space - MathOverflow
WEBThe key properties are that a subset of $\mathbb{R}^n$ is compact iff it is closed and bounded (Heine-Borel theorem) and that the image of a compact space under a …
Compact space - Wikipedia
WEBIn mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all …
Locally compact space - Encyclopedia of Mathematics
WEBApr 15, 2018 · A locally compact Hausdorff space $X$ is a completely-regular space. The partially ordered set of all its Hausdorff compactifications (cf. Compactification ) is a …
Countably compact space - Wikipedia
WEBIn mathematics a topological space is called countably compact if every countable open cover has a finite subcover.
Sequentially-compact space - Encyclopedia of Mathematics
WEBJun 22, 2014 · Sequentially-compact space. A topological space in which any infinite sequence of points contains a convergent subsequence (the Bolzano–Weierstrass …
Limit point compact - Wikipedia
WEBIn mathematics, a topological space is said to be limit point compact [1] [2] or weakly countably compact [3] if every infinite subset of has a limit point in . This property …
Compact Space | Brilliant Math & Science Wiki
WEBCompactness is a topological property that is fundamental in real analysis, algebraic geometry, and many other mathematical fields. In \ ( {\mathbb R}^n\) (with the standard …
Locally Compact -- from Wolfram MathWorld
WEBAug 22, 2024 · A topological space X is locally compact if every point has a neighborhood which is itself contained in a compact set. Many familiar topological spaces are locally …
Relatively compact subspace - Wikipedia
WEBIn mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) Y of a topological space X is a subset whose closure is compact.
Compact operator on Hilbert space - Wikipedia
WEBCompact operator on Hilbert space. In the mathematical discipline of functional analysis, the concept of a compact operator on Hilbert space is an extension of the concept of a …
Compactness - Encyclopedia of Mathematics
WEBNov 15, 2023 · A property which characterizes a wide class of topological spaces, requiring that from any covering of a space by open sets it is possible to extract a finite covering. …
What does it REALLY mean for a metric space to be compact?
WEBSep 2, 2015 · The discrete metric space on a finite set is compact. Closed bounded sets in $\mathbb{R}^n$ are compact. The discrete metric space on an infinite set is not …
Compact (German magazine) - Wikipedia
WEBCompact (stylized COMPACT, published with the subtitle "Magazine for Sovereignty" (German: Magazin für Souveränität) since October 2013) is a German media outlet, …