Glossary of Topology - L

L

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 x contains some member of B.

Local basis

See Local base.

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 local base consisting of compact neighbourhoods. 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 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.