On categorically related topologies and almost continuity

ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO Séria I: PRACE MATEMATYCZNE XXVIII (1989)

M

W

(Wroclaw)

On categorically related topologies and almost continuity

Abstract. Let / be a function on

into

where

are topological spaces, and let

be a second topology on

Under some assumptions,

*-continuity of / (or Baire property of / with respect to

implies that the set of continuity points of / is residual in

1. Introduction. This paper continues the author’s investigation of nearly continuous and almost continuous functions (cf. [11], [12], [13]) and extends in certain direction Fort’s study [3]. The main subject is to consider two topologies, the initial T and a second T* on a set Y, and to seek when T*- continuity (or some weaker property) of a function / on X into Y implies that its set of continuity points is residual in X.

In Section 2 we show that Basic Theorem of [3] is intrinsically connected with the notion of nearly continuity. If T has a countable network N, then the set C„(/, N) of points of nearly continuity with respect to N is residual in X (Lemma 1); if / is T *-continuous and condition (2) (on T, N, T*) holds, then C„(/, N) <= C (/) (Lemma 3), where C( f ) denotes the set of continuity points of /. Hence T*-continuity of / implies residuality of C( f ) provided T is of countable network weight and (2) holds (Theorem 1).

This yields the above-mentioned result of [3].

In Section 3 we give an analogon of Lemma 3, namely Lemma 4, which states that Ca(f, N) c: C( f ) whenever / has the Baire property with respect to T* and conditions (3)-(4) are satisfied (Ca(f, N) is the set of almost continuity points of / with respect to N). Lemmas 1 and 4 yield our central result: if / has the Baire property with respect to T*, N is countable and conditions (3H4) hold, then the set C (/) is residual in X (see Theorem 2 and Corollaries 5-10).

Both Lemma 3 and Lemma 4 can also be applied to homomorphisms or linear mappings which are nearly (almost) continuous under some category or barrelledness type assumptions on the spaces involved (see Proposition 2 and the final part of Section 2, Proposition 3 and Corollaries 2-4).

2. Categorically related topologies and nearly continuity. Let T and T* be

two topologies (in the sense of [2], i.e. families of open sets) on a set Y ; T is

388 Marek Wilhelm

said to be categorically related to T* if for every topological space X and every T*-continuous function / on X into Y the set C ( f ) of T-continuity points of/ is residual in X (i.e. X — C{f ) is of the first category in X). Fort [3] proved that T is categorically related to T* provided the following condition (a) holds. The pair T, T* (in this order) is said to satisfy condition (a) if there exist sequences {Aj}, \Kj) of subsets of Y such that:

(oq) Aj cz Kj for each j;

(a2) if y e V e T , then there exists j such that y e Aj c Kj cz V;

(a3) if y eAj , then there exists W e T* such that y e W and W — K j e T*.

Fort’s theorem has various interesting applications (thirteen theorems in [3])-

A family N of subsets of Y is called a network for T if for every V e T and every y e V there exists A e N such that y e A

V (cf. [2]). Condition (oc2) assures that both sequences [Aj], \Kj) are networks for T, which is thus of countable network weight.

Throughout the paper, we assume that / is a function on X into Y, T and T* are topologies on Y and N is a network for T. T is the initial topology on Y and operations, properties, etc., are considered with respect to T unless otherwise stated; in particular, C( f ) stands for the set of all continuity points of / with respect to T. Sometimes we consider only one topology on Y (namely T), as in the following passage ending with Corol­

lary 1 .

Let us define Cn( f N) (resp. Ca( f N)) to be the set of all points x e X such that for each A e N

J ( x ) e A implies x e l n t / " 1 (A) (resp. x e!ntZ ) ( / _1 (A))),

where D(E) (E cz X) denotes the set of all points x e X such that E is of the second category at x (i.e. E n U is of the second category in X for each neighbourhood U of x; cf. [ 6 ]). Since D(E) cz E for each E cz X, we have

Ca(f, N ) c C M N).

In the case where N is a base for T, Cn{f, N) (resp. Ca(f, N)) is precisely the set C„(f) (resp. Ca( f )) of all points of nearly continuity (resp. almost continuity) of / (cf. [11], [12], [13] and references therefrom). For an arbitrary network N the following inclusions hold:

Cn(f, N ) c: Cn(f) and C M N) cz C M )-

Le m m a

1 (cf. [12], Theorem 1). I f N is countable, then Ca{f, N

is residual in X.

P ro o f. The definition of Ca{f, N) implies the equality JV)= U r l ( A ) - \ x A D ( f - !(A)).

A e N

It is a consequence of the Banach category theorem that any set of the form E- l n t D( E) (E cz X) is of the first category in X (cf. [ 6 ], § 10). Hence X — Ca(f, N) is a countable union of first category sets.

The same assertion for Cn(f, N) does not involve the Banach category theorem, because E — In t£ is nowhere dense. We have always C( f ) c= Cn(f), and if X is a Baire space (i.e. X = D(X)), then C(f ) c- Ca(f). There exist simple conditions which imply the converse inclusions:

L

2. Let (Y, T) be a regular space. Consider the following two conditions on T, N and / :

(1J for each V e T and y e V there exists A e N such that y e A cz V and

(1„) for each V e T and y e V there exists A e N such that y e A c V and f - 1( A ) c : f - ^ V ) .

(1) If condition (la) holds, then Ca(f, N ) cz C(f).

(ii) If condition (1„) holds, then Cn{f, N) cz C(f).

P roof, (i) Let x e C a(f, N) and y = f ( x ) e V 0 eT. Let V e T satisfy y e V cz V cz V0 and choose A as in (1J. Then

x e l n t D ( / - 1 (4)) <= / “ 1 (К) c / “ 1 (v0), which shows that x e C( f ) . Part (ii) is similar.

The two lemmas taken together yield

Pr o p o s i t i o n

1. I f (Y, T) is a regular space, f : X -> Y and N is a countable network for T satisfying condition (1J, then the set C( f ) is residual in X.

Co r o l l a r y

1 ([3], Theorem

I f {Y, T) is a separable metrizable space and there exists a base В for T such that f ~ l (V) is closed for each VeB, then the set C( f ) is residual in X.

P ro o f. Let N cz В be a countable base for T ; condition (1„) (and hence ( 1 J) is satisfied.

All the above statements remain valid for multifunctions when continu­

ity and almost (nearly) continuity are replaced by lower semicontinuity and almost (nearly) lower semicontinuity (cf. [12], Theorem 1, and [13], Theorem 4). Now we are going to employ a second topology T * on Y.

L

3. Suppose that the following condition on T, N, T* is satisfied:

(2) for every V e T and y e V there exist A e N and W e T* such that y e A n W and Â* r \W czV (Â* denotes the T*-closure of A) I f f is T*- continuous, then C„ (/, N) cz C (/).

P roof. Assume, to get a contradiction, that xeC„(f , N) — C(f). There

exists V e T with y = f ( x ) E V and x<£Int / ~ 1 (F); x e / - 1 (Y— V). Choose

390 Marek Wilhelm

A, W as in (2). By the hypotheses, / i (W) is open and W —V cz W —A*, which gives

x s f ~ 1{ Y - V ) n f - l {W) c- f - x{ W - V ) cz f ~ 1{ W—A*).

Since x e C n{f, N), x e ln t f ~ l {A), and so

I n t / _ 1 (/4) n f ~ 1( W—A*) Ф 0 , which in turn implies

f ~ 1{ A ) n f ~ 1{ W- A* ) Ф 0 , because f ~ 1( W —Â*) is open. But this is impossible.

Lemmas 1 and 3 yield

Th e o r e m

1. Suppose T and T* are topologies on Y, N is a countable network for T and condition (2) is satisfied. Then for each space X and each T*-continuous function f : X - * Y the set C( f ) {of T-continuity points) is residual in X; in other words, T is then categorically related to T*.

R em ark 1. Suppose that the pair T, T* satisfies condition (a) with the appropriate sequences {Д,-}, {Kj}. Put N = {A}\. Then N is a countable network for T and the triple T, N, T* satisfies condition (2).

P ro o f. Given y e V e T , choose j and W as in (a2) and (a3). Let z e  J r \W . Since W —K j E T * and Aj r\{W — Kfi = 0 , z cannot be in

W —Kj. Hence z e W n K j c V .

According to Remark 1, Theorem 1 induces Basic Theorem of Fort [3].

Here is another consequence of Lemma 3:

Pr o p o s i t i o n

2. Let f be a function on X into Y, and let T, T* be topologies on Y. I f condition (2) with N = T is satisfied and f is nearly continuous and T*-continuous, then f is continuous (i.e. T-continuous).

P roof. C ( f ) z z C n( f , N) = Cn(f) = X.

Proposition 2 may be applied in several circumstances, because / is known to be nearly continuous under each of the following assumptions:

(i) X is a second category top. group, У is a ir-bounded top. group and / is a homomorphism (cf. [ 8 ], Lemma 2, and [12], Theorem 2; the notion of

<T-boundedness is recalled in the next section);

(ii) X is a second category top. vector space, У is a top. vector space and / is a linear mapping (cf. [ 8 ], Lemma 3, and [12], Theorem 3);

(iii) X and У are locally convex top. vector spaces, X is barrelled and / is linear (cf. [5], p. 57, and [13], Theorem 3);

(iv) X and У are locally convex vector lattices, X is order-quasibarrelled

and / is a lattice homomorphism (cf. [5], p. 123). (In (i) and (ii) / is even almost continuous — this will be used in Corollaries 3 and 4.) There are also other conditions of this type which guarantee nearly continuity of / ; see e.g.

[5], pp. 236-242. In each such situation / is continuous provided there exists a topology T * on Y (which needs not agree with the underlying group or vector space or lattice structure) such that (2) with N = T holds and / is T*- continuous.

3. Baire property and almost continuity. We are going to prove anal­

ogous of Lemma 3. Proposition 2 and Theorem 1 for functions which not necessarily are T*-continuous, but merely have the Baire property with respect to T*. In this section, S stands for a subfamily of T* (in applications S is a base or a subbase for T*); f is said to have the Baire property with respect to S if for every member W of S the inverse-image f ~ 1(W) has the Baire property in X. (E а X is said to have the Baire property if there exists an open set U cz X such that the symmetric difference E — U is of the first category in X.) We will also consider condition

(3) / - 1 (Ж) ^ D ( f ~ 1(W)) for each We S.

If / is almost T*-continuous, then / “ 1 {W) c In tD [ f ~ 1 (IT)) for all W e T*, and so (3) holds with S = T*.

R em ark 2. I f / i s a bijection, / -1 is T*-continuous and f ~ 1{W) is of the second category in X for each nonempty W e T*, then condition (3) with S = T* holds.

P ro o f. Let x e f ~ 1{ W) n U = : E, where W e T* and U is open in X.

Since / - 1 is T*-continuous, f (E) e T*, and consequently E is of the second category in X. Hence x E D ( f ~ 1(W)).

Le m m a

4. Assume the following two conditions on T, N, T* and S : (4) for each V e T and y E V there exists A e N such that у e A

Л*

V;

{5) for each A e N and y E Y — Â * there exists W e S such that y E W c Y —Â*.

I f f has the Baire property with respect to S and satisfies condition (3), then Ca {f, N) cz C (/).

P ro o f. Assume, to get a contradiction, that x ECa(f, N) — C(f). There exists V e T w i t h y = / ( x ) G K and x e X — In t / -1 (V) = / -1 (У— V). Choose A as in (4). Since x ECa(f, N) and y

A, x Elnt D ( f ~1 (A)) = : U. Hence there exists a point u e U r \ f ~ 1( Y —V) cz U n f ~ 1 (Y — Â*). By (5), there is W e S such that f ( u ) EW cz Y — Â*. Taking into account condition (3) we get

u e U n f - ' i W )

392 Marek Wilhelm

Thus U

1{W) is of the second category in X. Since A

Y — W,

U n f - i ( W ) ^ D ( f ~ 1( A ) ) n f - 1( W ) c : D ( X - f - l ( W ) ) - ( X - r 1(W)).

Hence the set on the right-hand side is of the second category as well, but this contradicts the assumption that has the Baire property (cf. [ 6 ],

§ 11.IV).

Not es, (i) If S is a base for T*, then (5) is trivially fulfilled (we will apply Lemma 4 to some special subbases S).

(ii) In case T = T*, (4) means regularity of T.

(iii) Condition (4) is stronger than (2) (take W = 7).

(iv) If (4) holds and / is T*-continuous, then condition (1„) is fulfilled ( Г 1{ А ) ^ г ' { Л * ) а Г Н П .

P

3. Let f be a function on X into Y, let T, T* be topologies on Y with T* <= T, and let S be a base for T*. Suppose that condition (4) with N

= T holds. I f f has the Baire property with respect to S and is almost continuous, then f is continuous.

Proof. Condition (3) is fulfilled because / is almost T*-continuous. By Lemma 4,

C( f ) = Ca{f, N ) ^{= Ca(f) =} x.

C

2 (cf. [12], Theorem 4). Let Y be a regular space with a base S. I f f : X -* Y has the Baire property with respect to S and is almost continuous, then f is continuous.

A topological group (У, •) is said to be a o-bounded group ([ 8 ]) if for every neighbourhood V of the neutral element there exists a sequence

00

!y„] <= Y such that Y = { J y nV v V y n; each separable group and each i

Lindelof group is cr-bounded. In connection with the next two corollaries, see points (i)— (ii) at the end of Section 2.

C

3. Let f be a homomorphism of a second category top. group X into a o-bounded top. group Y. I f there exists a coarser topology T* on Y (not necessarily a group topology) such that condition (4) with N — T holds and f has the Baire property with respect to T*, then f is continuous.

C

4 . Let f be a linear mapping of a second category top. vector space X into a top. vector space Y. I f there exists T* as above, then f is continuous.

In the special case where T* = T, Corollaries 3 and 4 yield Theorems 4

and 8 of Pettis [ 8 ].

T

2. Let f be a function on X into Y, let T, T * be topologies on Y, let N be a countable network for T, and let S be a base for T* (or at least a subfamily of T* satisfying (5)). Suppose also that condition (4) is fulfilled and f has the Baire property with respect to S.

(i) I f T* cz T, then there exists a residual Gs-set X 0 in X such that the restriction f \ X 0 is continuous.

(ii) I f condition (3) is satisfied, then the set C (f) (of T-continuity points) is residual in X.

Proof, (i) Assume first that A is a Baire space. By Lemma 1, the set Ca (/, N) is residual in X ; let X 0 be a dense G^-set contained in Ca (f, N) and put /о = f \ X 0. It is easy to verify that f 0 has the Baire property with respect to S and is almost continuous (cf. [6], § ll.V and § 10.IV). Thus f 0 satisfies condition (3) and Lemma 4 proves that C (/0) = X 0. If X is not a Baire space, then X' := In tD (X ) is a Baire space residual in X and f\X ' has the Baire property with respect to S (because X' is open); hence X 0 can be properly chosen in X'.

Part (ii) is a direct consequence of Lemmas 1 and 4.

When T* = T, Theorem 2 reduces itself to

C

5. Suppose Y is a regular space of countable network weight, S is a base for Y and f : X -> У has the Baire property with respect to S. Then there exists a residual Gs-set X 0 in X such that f \ X 0 is continuous. I f additionally condition (3) is satisfied, then the set C(f ) is residual in X.

The first assertion of Corollary 5 slightly generalizes Kuratowski’s characterization of functions having the Baire property and taking values in a second-countable space ([6], § 32.11). The second assertion should be compared with Corollary 2. Simple examples of real functions show that condition (3) plus Baire property does not imply continuity of / (e.g. step function) and that residuality of C( f ) does not imply condition (3) (e.g. x ~ l extended on R).

C

6. Let X be a Hausdorff space and let (Y, T*) be an analytic space (in the sense of [4] and [9]). Suppose that N is countable and condition (4) holds. I f the graph of f : X -> Y is a Souslin set in X x Y* (Y* is Y endowed with T*) and f satisfies condition (3) with S = T*, then C(f ) is residual in X.

P ro o f. In view of Theorem 2, it is sufficient to verify that / has the Baire property with respect to T*. But this is a well-known fact. (By Theorem of [9], for each T*-closed set F a Y the inverse-image f ~ 1(F) is a Souslin set in X, and so has the Baire property.)

C

7. Let X be a top. space, let Y be a separable dual Banach

space, let T(T*) be the norm (weak*) topology on Y, and let f : X -+Y possess

394 Marek W ilh elm

the Baire property with respect to T*. There exists a residual set X 0 such that f \ X 0 is continuous ; if additionally condition (3) with S'= T* holds, then the set

C( f ) (of norm continuity points) is residual in X.

Proof. By the Banach-Alaoglu theorem, each norm closed ball in Y is weakly* compact (see e.g. [10], 4.3); hence (4) holds with N being a countable base of open balls.

C

8 . Suppose X is a top. space, Y is a separable metrizable locally convex top. vector space, Z is the dual space of Y, S is the collection of all sets of the form { / e У : |z(y — y')| < e), where yeY^ z e Z , г > 0, and f : X —► Y possesses the Baire property with respect to S. Then there exists a residual set X 0 such that f \ X 0 is continuous; if (3) holds, then C(f ) is residual.

P roof. Let N be a countable base consisting of convex sets for the metric topology T on Y, and let T* be the weak topology on Y. It is well- known that closed convex sets are weakly closed (this implies (4)) and that condition (5) holds; cf. [10], 3.12 and 3.4.

C

9. Suppose X is a top. space, Y is a separable LP(m) space, where m is a finite measure on a а-field and 1 ^ p < oo, T* is the topology of convergence in measure (induced by the pseudonorm ||g||0 = j]g| л 1 dm), and f : X -* Y possesses the Baire property with respect to T*. Then f \ X 0 is continuous for some residual set X 0; if (3) with S = T* holds, then the set C(f ) (of Lp-norm continuity points) is residual in X.

P roof. Each norm closed ball in Y is T*-closed. (If К is the closed unit ball, gneK and H^-gfllo-+0, then the sequence \ди л\д\ v (~\g\)} с К converges in measure to g as well, and so the Lebesgue dominated con­

vergence theorem may be applied.)

Corollaries 8, 9 extend Theorems 10, 11 from [3].

Let h be a separately continuous function on X x P into Z, where X, P and Z are typological spaces. It is an old result of Baire that if X = P = Z

= R (the reals), then there exists a residual set X 0 in X such that X 0 x P c= C (h), i.e. h is jointly continuous at each point of X 0 x P. In [3] the same is proved for X arbitrary, P locally compact separable metrizable and Z separable metrizable, while in [1] even for X arbitrary, P second-countable and Z metrizable. We will show that in some cases the assumption on / can be slightly relaxed.

C

10. Suppose X is a top. space, P is a second-countable space, Z is a separable metrizable space and h: X x P ^ Z satisfies conditions

(6) for each x e X , h(x, •) is continuous;

(7) for each peP, h f , p) has the Baire property.

Suppose also that X is first-countable or P is locally compact.

Then there exists a residual G0-set X 0 in X such that h\X0 x P is continuous. I f additionally

(8) for every p e P and open W c^Z the inclusion E aD (E ) holds, where E = {x: h(x, p) eW \,

then X 0 x P c= C(h), where A'o is a residual Gô in X.

Proof. Let Y stand for the collection of all continuous functions on P into Z, let T be the compact-open and T* the point-open topologies on Y.

For E a P and f c Z put M{E, F) = {ye У: y(E) c F}. Let BP and Bz be countable bases for P and Z, respectively, and let N be the family of all sets M(V, W), where V eB P and W e B z ; N is a network for T (this remark constitutes Michael’s note [7]; see also [2], 3.4.G). All members of N are T*- closed, so that condition (4) holds. Let S consist of all sets M({p], W), where p e P and W e Bz ; S is a subbase for T* which satisfies condition (5). In view of (6), the equality f(x)(p) — h(x, p) defines a function / on X into Y.

Condition (7) implies that / has the Baire property with respect to S. By Theorem 2, there exists a residual G0-set I 0 in I such that f \ X 0 is continuous. Let К be a compact subspace of P; since T is the compact-open topology, we infer that h\X0 x K is continuous. Thus for P locally compact the first assertion is clear. If X is first-countable, x 0, xne X 0 and (x0, Po) elim(x„, p„), then the set {pn: n = 0, 1 ,2 ,...} is compact, and conse-

П

quently h(x0, p0) = lim h(x„, pn). Assume that (8) holds. Then condition (3) is П

satisfied, and Theorem 2 shows that C ( f) is residual. Since X is first- countable or P is locally compact, we infer (similarly as before) that h is jointly continuous at each point of C (/) x P, which concludes the proof.

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INSTITUTE OF MATHEMATICS, WROCLAW TECHNICAL UNIVERSITY WROCLAW, POLAND