Real maximal round filters in proximity spaces

Discrete Subsets of Proximity Spaces

“…We further observe that since the proximity class ir (8) contains the metric uniform structure which is not totally bounded, then it follows from results of Reed and Thron (cf. Corollary 2.1.3 of [12]) that ir (8) has at least c members. COROLLARY 3.5.…”

“…The following theorem shows that if (X, 8) is not Q-ciosed, then (X, °U) cannot be complete for all fyl t TT (8). In particular, (X, ô) cannot be realcomplete.…”

“…Let v 8 X be the real-completion of (X, b) (see [8], [11]). Then if card(5X -X) < 2 C , it follows that v §X = bX and ir(b) contains only the unique totally bounded uniform structure compatible with b.…”

“…Necessity is obvious. Conversely, if J^ is a maximal round filter in (X, 6), then PCX) -P*(X) implies that «F is real (see [8]). Since (X, 8) is realcomplete, J^ is fixed.…”