... Banach space. Thenany continuous map f : X → X for which theclosure of f(X) is compact must have a fixedpoint; that is, there is an x ∈ X with f(x)= x.In particular, any continuous mapping from ... regular, Tychonoff, and Urysohn spaces,refine and extend this list.separation by a continuous function Theproperty of a continuous function f : X →© 2001 by CRC Press LLCsieve of Eratosthenes[0, ... integers.analyticset The continuous image of a Borelset. More precisely, if X is a Polish space andA⊆X, thenA is analytic if there is a Borel setBcontained in a Polish space Y and a continuous f:X→Y...