one in which every two distinct points have disjoint neighbourhoods. Every Hausdorff space is T1. H-closed A space is H-closed, or Hausdorff closed or absolutely closed, if it is closed in every Hausdorff space containing it. Hereditarily P A space is hereditarily P for some property P if every subspace is also P. Hereditary A property of spaces is said to be hereditary if whenever a space has that property, then so does every subspace of it. For example, second-countability is a hereditary property. Homeomorphism If X and Y are spaces, a homeomorphism from X to Y is a bijective function f : X → Y such that f and f−1 are continuous. The spaces X and Y are then said to be homeomorphic. From the standpoint of topology, homeomorphic spaces are identical. Homogeneous A space X is homogeneous if, for every x and y in X, there is a homeomorphism f : X → X such that f(x) = y. Intuitively, the space looks the same at every point. Every topological group is homogeneous. Homotopic maps Two continuous maps f, g : X → Y are homotopic (in Y) if there is a continuous map H : X × [0, 1] → 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 maps. Hyper-connected A space is hyper-connected if no two non-empty open sets are disjoint Every hyper-connected space is connected. I Identification map See Quotient map. Identification space See Quotient space. Indiscrete space See Trivial topology. Infinite-dimensional topology See Hilbert manifold and Q-manifolds, i.e. (generalized) manifolds modelled on the Hilbert space and on the Hilbert cube respectively. Inner limiting set A Gδ set. Interior The interior of a set is the largest open set contained in the original set. It is equal to the union of all open sets contained in it. An element of the interior of a set S is an interior point of S. Interior point See Interior. Isolated point A point x is an isolated point if the singleton {x} is open. More generally, if S is a subset of a space X, and if x is a point of S, then x is an isolated point of S if {x}