Editing Glossary of general topology (section)

== L ==

;L-space: An ''L-space'' is a [[Hereditary property#In topology|hereditarily]] [[Lindelöf space]] which is not hereditarily [[Separable space|separable]].  A [[Suslin line]] would be an L-space.<ref name=GKW290>{{cite book | title=Sets and Extensions in the Twentieth Century | editor1-first=Dov M. | editor1-last=Gabbay | editor2-first=Akihiro | editor2-last=Kanamori | editor3-first=John Hayden | editor3-last=Woods | publisher=Elsevier | year=2012 | isbn=978-0444516213 | page=290 }}</ref>

;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 space|Lindelöf]]: A space is [[Lindelöf space|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 (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.<ref name=EGT65>Hart et al (2004) p.65</ref>

;[[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 space|Locally compact]]: A space is [[Locally compact space|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.<ref name=EGT65/>  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.<ref name=EGT65/>

; Locally dense: see ''Preopen''.

; [[Locally finite collection|Locally finite]]: A collection of subsets of a space is [[Locally finite collection|locally finite]] if every point has a neighbourhood which has nonempty intersection with only [[finite set|finite]]ly 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.<ref name=EGT65/>

;[[Locally path-connected]]: A space is [[locally path-connected]] if every point has a local base consisting of path-connected neighbourhoods.<ref name=EGT65/>  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 (topology)|Loop]]: If ''x'' is a point in a space ''X'', a [[loop (topology)|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]] ''S''<sup>1</sup> into ''X''.