The functor of continuous functions from compact CW-spaces to the reals - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T12:08:06Z http://mathoverflow.net/feeds/question/43974 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43974/the-functor-of-continuous-functions-from-compact-cw-spaces-to-the-reals The functor of continuous functions from compact CW-spaces to the reals roger123 2010-10-28T14:01:56Z 2010-10-28T15:55:13Z <p>The contravariant functor $C(-)$ given by $$\hom_{Top}(-,\mathbb{R}):cCW\to Rng$$ where $cCW$ is the category of compact CW complexes is injective on objects. What is known about surjectivity, faithfulness and fullness of this functor?</p> http://mathoverflow.net/questions/43974/the-functor-of-continuous-functions-from-compact-cw-spaces-to-the-reals/43989#43989 Answer by Dmitri Pavlov for The functor of continuous functions from compact CW-spaces to the reals Dmitri Pavlov 2010-10-28T15:53:01Z 2010-10-28T15:53:01Z <p>Corollary 4.1.(i) in Johnstone's book Stone Spaces (electronic version: <a href="http://gen.lib.rus.ec/get?nametype=orig&amp;md5=C26F62F69C32101307213F1960F85BA3" rel="nofollow">http://gen.lib.rus.ec/get?nametype=orig&amp;md5=C26F62F69C32101307213F1960F85BA3</a>) states that the category of realcompact spaces is dual to the full subcategory of the category of commutative rings consisting of rings of the form C(X). The functor C implements the duality.</p> <p>The category of compact CW-complexes embeds into the category of realcompact spaces as a full subcategory, hence the functor C is fully faithful.</p>