x contains some member of B. Local basis See Local base. Locally (P) space There are two definitions for a space to be "locally (P)" where (P) is a topological or set-theoretic property: that each point has a neighbourhood with property (P), or that every point has a neighourbood base for which each member has property (P). The first definition is usually taken for locally compact, countably compact, metrizable, separable, countable; the second for locally connected. Locally closed subset A subset of a topological space that is the intersection of an open and a closed subset. Equivalently, it is a relatively open subset of its closure. Locally compact A space is locally compact if every point has a compact neighbourhood: the alternative definition that each point has a local base consisting of compact neighbourhoods is sometimes used: these are equivalent for Hausdorff spaces. Every locally compact Hausdorff space is Tychonoff. Locally connected A space is locally connected if every point has a local base consisting of connected neighbourhoods. Locally dense see Preopen. Locally finite A collection of subsets of a space is locally finite if every point has a neighbourhood which has nonempty intersection with only finitely many of the subsets. See also countably locally finite, point finite. Locally metrizable/Locally metrisable A space is locally metrizable if every point has a metrizable neighbourhood. Locally path-connected A space is locally path-connected if every point has a local base consisting of path-connected neighbourhoods. A locally path-connected space is connected if and only if it is path-connected. Locally simply connected A space is locally simply connected if every point has a local base consisting of simply connected neighbourhoods. Loop If x is a point in a space X, a loop at x in X (or a loop in X with basepoint x) is a path f in X, such that f(0) = f(1) = x. Equivalently, a loop in X is a continuous map from the unit circle S1 into X. M Meagre If X is a space and A is a subset of X, then A is meagre in X (or of first category in X) if it is the countable union of nowhere dense sets. If A is not meagre in X, A is of second category in X. Metacompact A space is metacompact if every open cover has a point finite open refinement. Metric See Metric space. Metric invariant A