• Source: H-closed space

In mathematics, a Hausdorff space is said to be H-closed, or Hausdorff closed, or absolutely closed if it is closed in every Hausdorff space containing it as a subspace. This property is a generalization of compactness, since a compact subset of a Hausdorff space is closed. Thus, every compact Hausdorff space is H-closed. The notion of an H-closed space has been introduced in 1924 by P. Alexandroff and P. Urysohn.




Examples and equivalent formulations


The unit interval



[
0
,
1
]


{\displaystyle [0,1]}

, endowed with the smallest topology which refines the euclidean topology, and contains



Q
∩
[
0
,
1
]


{\displaystyle Q\cap [0,1]}

as an open set is H-closed but not compact.
Every regular Hausdorff H-closed space is compact.
A Hausdorff space is H-closed if and only if every open cover has a finite subfamily with dense union.




See also


Compact space




References


K.P. Hart, Jun-iti Nagata, J.E. Vaughan (editors), Encyclopedia of General Topology, Chapter d20 (by Jack Porter and Johannes Vermeer)

Kata Kunci Pencarian:

• Ichirō Nagai
• Nao Tamura
• Gigi Hadid
• Albert Einstein
• Bumi
• Universitas Victoria Manchester
• Derrick Hanson
• Universitas Jenderal Soedirman
• Wanita pada Abad Pertengahan
• Steam
• H-closed space
• Inner product space
• Normal space
• Glossary of general topology
• Hilbert space
• Banach space
• Closed manifold
• Topological vector space
• Closed system
• Clifford–Klein form