wa-realcompact

Last-modified: 2016-04-11 (月) 21:13:43

Definition

A Tychonoff space X is called wa-realcompact if for every imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20%5cbeta%20X%5csetminus%20X$%7d%25.png , there is a decreasing sequence imgtex.fcgi?%5bres=100%5d%7b$A_n$%7d%25.png of closed subsets in X which satisfies imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20%5ccap_%7bn%5cin%20%5comega%7d%20%5cmathrm%7bcl%7d_%7b%5cbeta%20X%7d%20A_n$%7d%25.png and imgtex.fcgi?%5bres=100%5d%7b$%5ccap_%7bn%5cin%20%5comega%7dA_n=%5cemptyset$%7d%25.png .

Remark

Let imgtex.fcgi?%5bres=100%5d%7b$X%5e*=%5cbeta%20X%5csetminus%20X$%7d%25.png . For imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20X%5e*$%7d%25.png , imgtex.fcgi?%5bres=100%5d%7b$%5cmathcal%7bF%7d%5ep$%7d%25.png ( imgtex.fcgi?%5bres=100%5d%7b$%5cmathcal%7bU%7d%5ep$%7d%25.png ) denotes the set of all free closed (resp. open) ultrafilters on X converging to p.

We devide imgtex.fcgi?%5bres=100%5d%7b$X%5e*$%7d%25.png into three domains;

imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bF%7d%280%29=%5c%7b%20p%5cin%20X%5e*:%5ctext%7bany%20%7d%5cmathcal%7bF%7d%5ep%5ctext%7b%20has%20ccip%20%7d%20%5c%7d%5c%5d%7d%25.png
imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bF%7d%28%5ctriangle%29=%5c%7b%20p%5cin%20X%5e*:%5ctext%7bno%20%7d%5cmathcal%7bF%7d%5ep%5ctext%7b%20has%20ccip%20%7d%20%5c%7d%5c%5d%7d%25.png
imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bF%7d%280%2c%5ctriangle%29=X%5e*%5csetminus%20%28%5cmathfrak%7bF%7d%280%29%5ccup%5cmathfrak%7bF%7d%28%5ctriangle%29%29%5c%5d%7d%25.png

Similarly, we introduce the following devision;

imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bU%7d%280%29=%5c%7b%20p%5cin%20X%5e*:%5ctext%7bany%20%7d%5cmathcal%7bU%7d%5ep%5ctext%7b%20has%20ccip%20%7d%20%5c%7d%5c%5d%7d%25.png
imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bU%7d%28%5ctriangle%29=%5c%7b%20p%5cin%20X%5e*:%5ctext%7bno%20%7d%5cmathcal%7bU%7d%5ep%5ctext%7b%20has%20ccip%20%7d%20%5c%7d%5c%5d%7d%25.png
imgtex.fcgi?%5bres=100%5d%7b%5c%5b%5cmathfrak%7bU%7d%280%2c%5ctriangle%29=X%5e*%5csetminus%20%28%5cmathfrak%7bU%7d%280%29%5ccup%5cmathfrak%7bU%7d%28%5ctriangle%29%29%5c%5d%7d%25.png

(cf. ccip).

Then generalization of realcompactness is characterized as following;

WA-realcompactness is introduced from this results so that

  • X is wa-realcompact iff imgtex.fcgi?%5bres=100%5d%7b$%5cmathfrak%7bF%7d%280%29=%5cemptyset$%7d%25.png .

See [Isikawa] for details.

Reference

T. Isiwata, Closed ultrafilters and realcompactness, Pacific J. Math. 94 (1981) 68-71.