i c-space -I
Gra i -@rage [2] has introduced two inteiesting generalizations of first countable amely W-spaces and w-spaces. l'hese spacer dre defined in terms of a n infinite game. The purpose of this note 13 to give simple and purely 1 conditions characterizing these spaces. It is foui;d that, in practice, the king conditions we give are far easier to use than the defining conditions of Gruenhage. For example, Galvin [I] has found the characterization of w-spaces given here to be very useful.
Let p be a point in ~1 topological space X and consider the following twa-person infinite game at p: Player I chooses an open neighborhood U1 of 2, and then player II chooses a point xi E Ui; p!ayer I then chooses another open set U2 containing p, player II chooses some point X~E U2, and so on. Player I wigls the game if the sequence (x,) converges to p. The precise definition of the game is as follows.
topological space and take p E X. A strategy at p for player I is a 4Vp, where 9(X) is the set of all finitz sequences in X and ?$, is the set of all open neighborhoods of p in X. A strategy at p for pkzyer II is a function T : 9(X) x NP +X such that T(F, U)E U for each F E 9(X). I et p E X be given and let a and r be strategies at p for players I and II respectively. The:n 0 and r together determine a unique ssqueltce (x,) as follows:
where ($3) means the empty sequence.