compact support

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 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\}.

Revised on June 5, 2012 02:41:02 by Urs Schreiber (