Charles University in Prague, Faculty of Mathematics and Physics
Abstract
summary:We prove that a Hausdorff space X is locally compact if and only if its topology coincides with the weak topology induced by C∞(X). It is shown that for a Hausdorff space X, there exists a locally compact Hausdorff space Y such that C∞(X)≅C∞(Y). It is also shown that for locally compact spaces X and Y, C∞(X)≅C∞(Y) if and only if X≅Y. Prime ideals in C∞(X) are uniquely represented by a class of prime ideals in C∗(X). ∞-compact spaces are introduced and it turns out that a locally compact space X is ∞-compact if and only if every prime ideal in C∞(X) is fixed. The existence of the smallest ∞-compact space in βX containing a given space X is proved. Finally some relations between topological properties of the space X and algebraic properties of the ring C∞(X) are investigated. For example we have shown that C∞(X) is a regular ring if and only if X is an ∞-compact P∞-space