General topology Basic notions: open sets, neighbourhoods, bases Concept
Closed set
Summary
A closed set in a topological space is a set whose complement is open. In a metric space, the closed sets are the sets that contain the limits of all their convergent sequences.
Where this leads
81 pages build on this one.
- Dynamic equation on time scales
- Vietoris topology
- Montel space
- Remmert–Stein theorem
- Urysohn space
- Complemented subspace
- Closed subgroup theorem
- Michael selection theorem
- Cartan–Hadamard theorem
- Sorgenfrey plane
- Perfect set
- Portmanteau theorem
- Mumford–Shah functional
- Céa's lemma
- Cartan's coherence theorem for ideal sheaves
- Adjunction space
- Chow's theorem
- Preissmann's theorem
- Generic point
- Hahn–Banach separation theorem
- Dimension of an algebraic variety
- Barrelled space
- Absolute neighborhood retract
- Tychonoff space
- Kakutani fixed-point theorem
- Bishop's theorem on analytic sets
- Lax–Milgram theorem
- Excluded point topology
- Clopen set
- Germ of an analytic set
- Hausdorff topological group
- Minimum total potential energy principle
- Quotient of a normed space
- Proper cone
- Varadhan's lemma
- Dimension of an analytic set
- Time-scale calculus
- Hochster's theorem on spectral spaces
- Closed immersion
- Cartan's theorem on continuous homomorphisms
- Hilbert space adjoint
- Open subgroups are closed
- Supporting hyperplane theorem
- Perfectly normal space
- Irreducible analytic germ
- Zariski topology
- Straszewicz's theorem
- Kuratowski closure axioms
- Sequential space
- Closed graph theorem
- Lions–Stampacchia theorem
- Smith–Volterra–Cantor set
- Hilbert projection theorem
- Regular and singular points of an analytic set
- Local parametrization theorem
- Noetherian topological space
- Large deviations theory
- Galerkin method
- Sober space
- Minkowski–Carathéodory theorem
- Sierpiński space
- Closed operator
- Krein–Smulian theorem
- Hopf–Rinow theorem
- Signorini problem
- Sanov's theorem
- Dugundji extension theorem
- Whitney extension theorem
- Cramér's theorem
- Tessellation
- Moore plane
- Ladyzhenskaya–Babuška–Brezzi condition
- Delta derivative on a time scale
- Inductive dimension
- Riesz representation theorem
- Complex analytic variety
- Rate function
- Matrix Lie group
- F-sigma set
- Irreducible component
- Hellinger–Toeplitz theorem
Further reading
- Web https://en.wikipedia.org/wiki/Closed_set, lead section
About this page
| Tag | 0ZCK8DQYN1 |
|---|---|
| URI | m2://0ZCK8DQYN1 |
| Kind | concept |
| Writing | W1 stated |
| MSC | 54A05 |
| Same as |
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What this page needs for W2
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What links here
Definitions
Used by
- Vietoris topology
- Montel space
- Urysohn space
- Complemented subspace
- Closed subgroup theorem
- Michael selection theorem
- Sorgenfrey plane
- Perfect set
- Portmanteau theorem
- Mumford–Shah functional
- Adjunction space
- Generic point
- Hahn–Banach separation theorem
- Dimension of an algebraic variety
- Barrelled space
- Absolute neighborhood retract
- Tychonoff space
- Kakutani fixed-point theorem
- Excluded point topology
- Clopen set
- Hausdorff topological group
- Quotient of a normed space
- Proper cone
- Time-scale calculus
- Hochster's theorem on spectral spaces
- Closed immersion
- Open subgroups are closed
- Perfectly normal space
- Zariski topology
- Kuratowski closure axioms
- Sequential space
- Closed graph theorem
- Lions–Stampacchia theorem
- Smith–Volterra–Cantor set
- Hilbert projection theorem
- Noetherian topological space
- Large deviations theory
- Sober space
- Sierpiński space
- Closed operator
- Krein–Smulian theorem
- Hopf–Rinow theorem
- Dugundji extension theorem
- Whitney extension theorem
- Tessellation
- Moore plane
- Inductive dimension
- Complex analytic variety
- Rate function
- Matrix Lie group
- F-sigma set
- Irreducible component