Editing Glossary of general topology (section)

== H ==

; [[Hausdorff space|Hausdorff]]: A [[Hausdorff space]] (or '''[[T2 space|T<sub>2</sub>]] space''') is one in which every two distinct points have [[Disjoint sets|disjoint]] neighbourhoods. Every Hausdorff space is T<sub>1</sub>.
; [[H-closed space|H-closed]]: A space is H-closed, or '''Hausdorff closed''' or '''absolutely closed''', if it is closed in every Hausdorff space containing it.
; [[Hemicompact space|Hemicompact]]: A space is hemicompact, if there is a sequence of compact subsets so that every compact subset is contained in one of them.
; Hereditarily ''P'': A space is hereditarily ''P'' for some property ''P'' if every subspace is also ''P''.
; [[Hereditary property|Hereditary]]: A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it.<ref name=ss4>Steen &amp; Seebach p.4</ref> For example, second-countability is a hereditary property.
; [[Homeomorphism]]: If ''X'' and ''Y'' are spaces, a [[homeomorphism]] from ''X'' to ''Y'' is a [[bijection|bijective]] function ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' such that ''f'' and ''f''<sup>−1</sup> are continuous. The spaces ''X'' and ''Y'' are then said to be '''homeomorphic'''. From the standpoint of topology, homeomorphic spaces are identical.
; [[Homogeneous space|Homogeneous]]: A space ''X'' is [[Homogeneous space|homogeneous]] if, for every ''x'' and ''y'' in ''X'', there is a homeomorphism ''f'' : ''X'' &nbsp;→&nbsp; ''X'' such that ''f''(''x'') = ''y''. Intuitively, the space looks the same at every point. Every [[topological group]] is homogeneous.
; [[homotopic|Homotopic maps]]: Two continuous maps  ''f'', ''g'' : ''X'' &nbsp;→&nbsp; ''Y'' are [[homotopic]] (in ''Y'') if there is a continuous map ''H'' : ''X'' × [0, 1] &nbsp;→&nbsp; ''Y'' such that ''H''(''x'', 0) = ''f''(''x'') and ''H''(''x'', 1) = ''g''(''x'') for all ''x'' in ''X''. Here, ''X'' × [0, 1] is given the product topology. The function ''H'' is called a '''homotopy''' (in ''Y'') between ''f'' and ''g''.
; Homotopy: See '''[[homotopic|Homotopic maps]]'''.
; [[Hyperconnected space|Hyperconnected]]: A space is hyperconnected if no two non-empty open sets are disjoint<ref name=ss29/>  Every hyperconnected space is connected.<ref name=ss29/>