Limits in function spaces and compact groups
by
Joan Hart
University of Wisconsin Oshkosh
Coauthors: Kenneth Kunen

Suppose that g is a limit point of <f_n : n in \omega> in C_p(X, Y). Even when X and Y are both compact metric spaces, there need not be an infinite B subset or equal \omega such that <f_n : n in B> converges pointwise to g. We show that if X is compact and Y metric, then there is a filter F subset [\omega]^\omega such that for each x in X there is a B in F such that <f_n(x) : n in B> --> g(x).

For X a topological group with identity 1, we consider the sequence <xⁿ : n in \omega>. For B in [\omega]^\omega, let C^X_B be the set of all x in X such that <xⁿ : n in B> converges to 1, and for F subset [\omega]^\omega a filter, let D^X_F = \cup {C^X_B : B in F}. For X compact metric, the C_p result yields an F such that D^X_F = X.

For X any non-trivial compact group, C^X_B = X for some B in [\omega]^\omega iff X is totally disconnected. Moreover, we describe a Borel F subset or equal P(\omega), such that when X is abelian and not totally disconnected, D^X_F is a Haar null subgroup not contained in any C^X_B.

Date received: May 30, 2003