nLab compact support




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory



A function f:XVf\colon X \to V on a topological space with values in a vector space VV (or really any pointed set with the basepoint called 00) has compact support (or is compactly supported) if the closure of its support, the set of points where it is non-zero, is a compact subset. That is, the subset f 1(V{0})¯\overline{f^{-1}(V \setminus \{0\})} is a compact subset of XX.

Typically, XX is Hausdorff, ff is a continuous function, and VV is a Hausdorff topological vector space (or at least a pointed topological space whose basepoint is closed), so that f 1(V{0})f^{-1}(V \setminus \{0\}) is an open subspace of XX, yet any compact subspace of XX must be closed; this is why we take the closure.

If we work with locales instead of topological spaces, then a closed point 00 in VV still has an open subspace of VV as its formal dual, and we use this in the place of V{0}V \setminus \{0\}.

Last revised on April 11, 2018 at 09:42:39. See the history of this page for a list of all contributions to it.