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

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