Compact space
Definition
A topological space is compact when each open cover of has a finite subcover.
Remarks
Some authors, mainly in the French tradition, also require a compact space to be Hausdorff, and they call the general notion quasi-compact. Each finite topological space is compact. An infinite set with the discrete topology is not compact. For metric spaces, compactness is the same as sequential compactness, by this theorem. The continuous image of a compact space is compact.
Explanations
L1 human
In a compact space, each way to cover the space by open sets can be reduced to finitely many of them. This finiteness lets you turn local facts, which hold near each point, into global facts with one bound for the whole space.
What this node needs for W2
- edge not_instance_of in: one non-example that fails one clause (0 of 1)
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What depends on this
32 nodes use this node, directly or through other nodes.
- Finite intersection property 00BWV5CKCF
- Compact set 00R0X1WGPX
- Arzelà–Ascoli theorem 0381S8Z5P0
- Equivalence of norms in finite dimension 03MC70ZRNR
- Torus 03NM7DZXXC
- Lebesgue number lemma 040CFNJEAP
- Continuous bijection from a compact space to a Hausdorff space is a homeomorphism 0430ETCMKD
- Compact iff complete and totally bounded 054Q5PW5GX
- Closed subsets of compact sets are compact 066HW6D9A7
- Heine–Cantor theorem 06B8BNJCD6
- Heine–Borel theorem 07PFYV65MB
- Sequence spaces lp 08P6T2WH6R
- Product of compact sets is compact 09ZKE5XQ74
- Finite-dimensional normed spaces are complete 0ABP2F2ZP7
- Linear maps on finite-dimensional spaces are continuous 0C13E623WD
- Locally compact space 0EQMWQ4WMJ
- Cantor's intersection theorem 0EZBG8SKJM
- Tychonoff's theorem 0JTRSJKCFB
- Compactness and sequential compactness agree in metric spaces 0K8SGQSV7G
- Continuous image of a compact set is compact 0K9AAJ8G51
- p-norm on R^n 0KNJWEVSMG
- One-point compactification 0MM45BT6HC
- Compact Hausdorff spaces are normal 0MQQQ5NA2S
- Compact subsets of a Hausdorff space are closed 0MYFJVMN7M
- Compactness via open covers 0P6PMJB4MA
- Heine–Borel theorem in finite-dimensional normed spaces 0PHT6NBYPB
- Compact subsets of a metric space are bounded 0R7BTEWCJ2
- Cantor set is homeomorphic to the space of binary sequences 0RQ8DHHR8V
- Continuous real function on a compact set attains its bounds 0V4DET59NW
- Compact closed unit ball implies finite dimension 0VZVV8VME5
- Circle as a quotient of the real line 0WH1GVBJNH
- Proper metric space 0Z4V98VN5E
Sources
- Web https://en.wikipedia.org/wiki/Compact_space, Open cover definition
What links here
Used by
- Compact set 00R0X1WGPX
- Arzelà–Ascoli theorem 0381S8Z5P0
- Lebesgue number lemma 040CFNJEAP
- Heine–Cantor theorem 06B8BNJCD6
- Tychonoff's theorem 0JTRSJKCFB
- Compact Hausdorff spaces are normal 0MQQQ5NA2S