I. FIRST CLASS FUNCTIONS AROUND A SUBS£T OF A POLISH SPACE
A, Characterization of first class functions around sets:
Let D be a subset of a completely regular topological space K
and let f be a real-valued function defined on K. We shall say
that f is in the first class around D if for every non-empty
subset F of D and any e 0, there exists an open set 0 in K
such that 0 ( 1 F * 0 and the oscillation of f on 0 f l F is less
than e. We shall denote by B,(K,D) the family of such functions
endowed with the topology of pointwise convergence on K. Note that
if K is Polish B^K) = B^K/K) is the well known class of Baire-1
functions on K.
Recall that a filter U on the set of real-valued functions on
K is said to converge quasi-uniformly on K to the function f if
for all e 0 and each non-empty closed subset F of K, there
exists an open subset 0 of K with 0 D F * 0 and a set A in U
such that sup f(x) - g(x) I j e.
xe0riF,g€A
Remark: This notion of convergence is behind the remarkable
"subsequence principles" obtained in [5] for subsets of B,(K).
Note that if K is a Polish space, then a sequence of continuous
functions (f ) pointwise converges to f on K if and only if it
converges quasi-uniformly on K to f. Indeed, for each e0 and
6 £
every closed subset F of K we have F » U F where F is the
I
closed set n {x F; f (x) - f
(x)I e}.« B"
y the Baire category
V n ra
m n
theorem, at least one of the F , has a non-empty interior. This
clearly proves our claim.
The notion of convergence that is compatible with the class B.(K,D)
is the following:
We shall say that a filter U on the set of real-valued
functions on K converges quasi-uniformly around D to the function
£ if for all e 0 and for any non-empty subset F of D, there
exists an open subset 0 of K with 0 ( 1 F * 0 and a set A in U
such that sup f(x) - g(x) ^ e.
x€0flF,g€A
We denote by osc(f L) the oscillation of f on a subset L of
K and for each x in K we will write
osc(f)(x) = inf{osc(f U);U open in K containing x}.
The set C(f) = {x K;osc(f)(x) = 0} is then the set of points
6
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