Editing Glossary of general topology (section)

== Q ==

;Quasicompact: See '''[[compact space|compact]]'''. Some authors define "compact" to include the [[Hausdorff space|Hausdorff]] separation axiom, and they use the term '''quasicompact''' to mean what we call in this glossary simply "compact" (without the Hausdorff axiom). This convention is most commonly found in French, and branches of mathematics heavily influenced by the French.

;[[Quotient map (topology)|Quotient map]]: If ''X'' and ''Y'' are spaces, and if ''f'' is a [[surjection]] from ''X'' to ''Y'', then ''f'' is a quotient map (or '''identification map''') if, for every subset ''U'' of ''Y'', ''U'' is open in ''Y'' [[if and only if]] ''f''<sup>&nbsp;{{mono|-}}1</sup>(''U'') is open in ''X''.  In other words, ''Y'' has the ''f''-strong topology.  Equivalently, <math>f</math> is a quotient map if and only if it is the transfinite composition of maps <math>X\rightarrow X/Z</math>, where <math>Z\subset X</math> is a subset. Note that this does not imply that ''f'' is an open function.

;[[Quotient space (topology)|Quotient space]]: If ''X'' is a space, ''Y'' is a set, and ''f'' :&nbsp;''X''&nbsp;→&nbsp;''Y'' is any [[surjection|surjective]] function, then the [[Quotient topology]] on ''Y'' induced by ''f'' is the finest topology for which ''f'' is continuous. The space ''X'' is a quotient space or '''identification space'''. By definition, ''f'' is a quotient map. The most common example of this is to consider an [[equivalence relation]] on ''X'', with ''Y'' the set of [[equivalence class]]es and ''f'' the natural projection map. This construction is dual to the construction of the subspace topology.