Editing Glossary of general topology (section)

== R ==

; Refinement: A cover ''K'' is a [[refinement (topology)|refinement]] of a cover ''L'' if every member of ''K'' is a subset of some member of ''L''.
; [[Regular space|Regular]]: A space is [[regular space|regular]] if, whenever ''C'' is a closed set and ''x'' is a point not in ''C'', then ''C'' and ''x'' have [[Disjoint sets|disjoint]] neighbourhoods.
; [[T3 space|Regular Hausdorff]]: A space is [[T3 space|regular Hausdorff]] (or '''T<sub>3</sub>''') if it is a regular T<sub>0</sub> space.  (A regular space is Hausdorff [[if and only if]] it is T<sub>0</sub>, so the terminology is consistent.)
; {{visible anchor|[[Regular open set|Regular open]]}}: A subset of a space ''X'' is regular open if it equals the interior of its closure; dually, a regular closed set is equal to the closure of its interior.<ref name="Steen & Seebach 1978 p.6">Steen & Seebach (1978) p.6</ref>  An example of a non-regular open set is the set  ''U'' = {{open-open|0,1}} ∪ {{open-open|1,2}} in '''R''' with its normal topology, since 1 is in the interior of the closure of ''U'', but not in ''U''. The regular open subsets of a space form a [[complete Boolean algebra]].<ref name="Steen & Seebach 1978 p.6"/>
; [[Relatively compact]]: A subset ''Y'' of a space ''X'' is [[relatively compact]] in ''X'' if the closure of ''Y'' in ''X'' is compact.
; Residual: If ''X'' is a space and ''A'' is a subset of ''X'', then ''A'' is residual in ''X'' if the complement of ''A'' is meagre in ''X''. Also called '''comeagre''' or '''comeager'''.
; Resolvable: A [[topological space]] is called [[resolvable space|resolvable]] if it is expressible as the union of two [[disjoint sets|disjoint]] [[dense subset]]s.
; Rim-compact: A space is rim-compact if it has a base of open sets whose boundaries are compact.