is open in the subspace topology on S. Isometric isomorphism If M1 and M2 are metric spaces, an isometric isomorphism from M1 to M2 is a bijective isometry f : M1 → M2. The metric spaces are then said to be isometrically isomorphic. From the standpoint of metric space theory, isometrically isomorphic spaces are identical. Isometry If (M1, d1) and (M2, d2) are metric spaces, an isometry from M1 to M2 is a function f : M1 → M2 such that d2(f(x), f(y)) = d1(x, y) for all x, y in M1. Every isometry is injective, although not every isometry is surjective. K Kolmogorov axiom See T0. Kuratowski closure axioms The Kuratowski closure axioms is a set of axioms satisfied by the function which takes each subset of X to its closure: Isotonicity: Every set is contained in its closure. Idempotence: The closure of the closure of a set is equal to the closure of that set. Preservation of binary unions: The closure of the union of two sets is the union of their closures. Preservation of nullary unions: The closure of the empty set is empty. If c is a function from the power set of X to itself, then c is a closure operator if it satisfies the Kuratowski closure axioms. The Kuratowski closure axioms can then be used to define a topology on X by declaring the closed sets to be the fixed points of this operator, i.e. a set A is closed if and only if c(A) = A. Kolmogorov topology TKol = {R, }∪{(a,∞): a is real number}; the pair (R,TKol) is named Kolmogorov Straight. L L-space An L-space is a hereditarily Lindelöf space which is not hereditarily separable. A Suslin line would be an L-space. Larger topology See Finer topology. Limit point A point x in a space X is a limit point of a subset S if every open set containing x also contains a point of S other than x itself. This is equivalent to requiring that every neighbourhood of x contains a point of S other than x itself. Limit point compact See Weakly countably compact. Lindelöf A space is Lindelöf if every open cover has a countable subcover. Local base A set B of neighbourhoods of a point x of a space X is a local base (or local basis, neighbourhood base, neighbourhood basis) at x if every neighbourhood of