Some observations on mildly Menger property and topological games

Abstract

In this paper, we defined a new game namely compact-clopen game. In
a zero-dimensional space, the game Gfin(O;O) is equivalent to Gfin(CO; CO)
and compact-open game is equivalent to compact-clopen game. An example
is given for a space on which the game Gfin(CO; CO) is undetermined. Also we
introduced a new game namely K-quasi-component-clopen game and proved
that this game is equivalent to compact-clopen game. Then we proved that
if X is an union of countably many quasi-components of compact sets, then
TWO has a winning strategy in the game Gfin(CO; CO).