*Definition [#f4138fb1]
A Tychonoff space X is called wa-realcompact if for every &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20%5cbeta%20X%5csetminus%20X$%7d%25.png);, there is a decreasing sequence &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$A_n$%7d%25.png); of closed subsets in X which satisfies &ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5ccap_%7bn%5cin%20%5comega%7dA_n=%5cemptyset$%7d%25.png);.

*Remark [#t20fdf4f]
Let &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$X%5e*=%5cbeta%20X%5csetminus%20X$%7d%25.png);. For &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$p%5cin%20X%5e*$%7d%25.png);, &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathcal%7bF%7d%5ep$%7d%25.png); (&ref(http://www.eaflux.com/imgtex/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 &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$X%5e*$%7d%25.png); into three domains;
#ref(http://www.eaflux.com/imgtex/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);
#ref(http://www.eaflux.com/imgtex/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);
#ref(http://www.eaflux.com/imgtex/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;
#ref(http://www.eaflux.com/imgtex/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);
#ref(http://www.eaflux.com/imgtex/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);
#ref(http://www.eaflux.com/imgtex/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;
-X is [[almost realcompact]] iff &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathfrak%7bU%7d%280%29%5ccup%5cmathfrak%7bU%7d%280%2c%5ctriangle%29=%5cemptyset$%7d%25.png);;
-X is [[c-realcompact]] iff &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathfrak%7bU%7d%280%29=%5cemptyset$%7d%25.png);;
-X is [[a-realcompact]] iff &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathfrak%7bF%7d%280%29%5ccup%5cmathfrak%7bF%7d%280%2c%5ctriangle%29=%5cemptyset$%7d%25.png);.

WA-realcompactness is introduced from this results so that
-X is wa-realcompact iff &ref(http://www.eaflux.com/imgtex/imgtex.fcgi?%5bres=100%5d%7b$%5cmathfrak%7bF%7d%280%29=%5cemptyset$%7d%25.png);.

See [Isikawa] for details.

*Reference [#c5debcb1]
T. Isiwata, ''Closed ultrafilters and realcompactness'', Pacific J. Math. 94 (1981) 68-71.