first category function spaces under the topology of pointwise ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 50, July 1975

FIRST CATEGORY FUNCTION SPACES UNDER THE TOPOLOGY OF POINTWISE CONVERGENCE R. A. McCOY ABSTRACT. wise

A large class

convergence,

The question

category sets)

(i.e.,

seems

space

are Baire

ever,

an example

the supremum

topology spaces

spaces

changes

of real-valued

The notation from

topology

is also which

is of first

interval,

into

If the domain

that the Y, under

space

is

a large

class

are Baire

this

/, with

of function

spaces.

However,

of pointwise

topology, domain in this

converg-

the function and range paper

that the

the topology

of point-

category.

Y under

is generated

unit

nonpathological

on

How-

Y such

from the Theorem

C (X, Y) will stand X into

Under

functions

category).

space

when the topology

for most

there

with the compact-open

topology

it will follow continuous

convergence,

functions This

category

For example,

theorem, topology

Baire

category.

agrees

spaces.

is completely

metric

is of first

sub-

on the function

by the Baire category

so there

dramatically

on the function

are of first

spaces.

space,

dense

space

the supremum

I, the closed

topology

is of first

of nowhere

if the range

subspace

is of first

metric

the compact-open

is imposed

Thus having

from

topology,

on the function

then

no open

of point-

functions

union

in L4] of a metrizable

functions

metric

having

ence

(i.e.,

is given

the supremum

the situation

space.

topology

When the topology

topology,

spaces

the

of continuous

unanswered.

metric

spaces

of continuous

compact,

a space

under

category.

as a countable

of function

which

space

as to when

to be relatively

class

spaces,

to be of first

so is the function

is a large

space

of function

shown

can be written

is the supremum

metrizable,

wise

are

for the space

the topology

of all continuous

of pointwise

convergence.

by the base

$ = I H L*;. V.]|x. e X and V. is open in YK Presented

to the Society,

AMS (MOS) subject

January

classifications

Keywords and phrases. Spaces topology of pointwise convergence.

23, 1975; received by the editors April 13, 1974.

(1970). Primary 54C35; Secondary of first

category,

Baire

spaces,

54D99.

function

spaces,

Copyright © 1975, American Mathematical Society License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

431

432

R. A. McCOY

where

each

Lx, V] = {f£ CpiX,

the topology subspace

inherited

from the product

of ^xeX Yx, where

Because work with

of the nature

basic

open

sets

use a characterization

the proof of this

open

contains

subsets

a member

than

of this

arbitrary

it is much easier

sets.

category

For this

given

each

to

reason

in [5].

[6],

Similar

in [2] and L3J•

a pseudo-base that

C (X, Y) as a

is due to Oxtoby spaces

as

Y.

open

of first

for Baire

space,

of

on C (X, y),

of spaces

of X such

be considered

considering

is a copy

characterization

can be found

If X is a topological

Y

rather

Basically,

nonempty

each

topology,

of the topology

we shall

characterizations

Y) | fix) £ V\. It can also

for

X is a collection

nonempty

open

subset

of of X

collection.

Let ? be a pseudo-base for X. Define S[X, 9] = {/: 9 -» 9 | fill) C U for every U £9\. If Ue9 and f,g£S[X,9], then define

iu. f, g)l = giu), and for i > 1,

ÍfÜU. f, £>,_,), if i is even, gi Y such

X is completely completely

subset

Hausdorff

Hausdorff

Theorem.

Let

open that

fix)

of

X contain

respect

to Y if for

of X and every

Y, there

exists

finite

set

a continuous

i = 1, ■■■, n. Note that

to the reals

if and only

if it is

sense.

a convergent

X be completely

three

with

£ V. fot every

with respect

any of the following

of first

points

subsets

in the usual

of X, and let

satisfies

Hausdorff

■■■, x \ of distinct

\V, , • • • , V i of nonempty function

X completely

sequence

Hausdorff

conditions,

with

which

is infinite

respect

as a

to Y. If Y

then the space

C iX, Y) is

category. (i)

There

exist

two nonempty

open

subsets

of

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Y with

disjoint

closures.

433

FIRST CATEGORY FUNCTION SPACES (ii) such

There

that,

quence

exists

for each

a sequence

sequence

\W.} of nonempty

\y.\

in Y with

y.£

open subsets

W. for every

of Y

i, no subse-

of \y .\ converges.

(Hi) Y is of first category. Proof. subset

Let

\t.\

be a convergent

of X. To prove

disjoint

closures.

l/VCuj

| either

part

Let

a> denote

N is finite

i/V^i be an indexing

sequence

(i), let

U and

in

the set of positive

or cu\/V is finitei.

of R on tu. For each

spect

open

to

subset

of C (X, Y) since

Y. Now suppose

of

integers,

n,

i£a>,

let

• Let

as a

Y with

and let

R =

so let

Wl = U if i£ N^ and

^n = U^l^i'

X is completely

/ef)0"-!*

is infinite

subsets

Now R is countable,

let W^ = V if i 4 Nn. For each n £ to, define a dense

X which

V be open

W*J> which is Hausdorff

M, = {z e a> \ fit)

with re-

e U\, and let

M2 = \i£ cu I /G.) e V\. Suppose that M1 is finite, so that oAzMj = zVfefor some

¿ecu.

Now f£ W

so that for some

¿ecu,

f(t.)£W,.

If z'eMp

then

fit) £ V, so that i 4 My But if i 4 Mp then fit) e U, so that i £ My Either

way is a contradiction,

cannot

be finite.

But then

continuity ' of 'f. Therefore category

in this

To prove

above,

each

O00 . Wn = ^0,

so that

lW.| be a sequence

sequence

element

with respect

[y.i

to

in

CAX, P

since

subsequence

n°ll^'

9> is a pseudo-base

Let V = f|"=i

positive

For each

of \t.\,

integer

¿ecu

let

x. = z1 ,,,,

it is convergent.

of \W.}, {cpix.)\

would

not converge,

for

CAX,

of

i £ tu, no Y) defined

Haus-

for G AX, Y). Define

that

tmi\/\

,

.

But if were

f°r every

which

open subsets

E*f, V.]£9> be arbitrary.

such

/' g)' then ],

M2

contradicts

W. for every

Now if i> is the base

Y. Hence

the smallest

y.e

Now let xn +.,l = tm{V „,, ) and Vn ++1i = Wm\VM/v) Then define S[C(X,

Similarly

which

of nonempty

Y with

of 9> is nonempty

f£ S[C (X, Y), $] as follows. z?z(V) denote

be finite.

not converge,

' ' n=l

of iy-S converges.

then

dorff

let

for each

subsequence

Alj cannot

could

case.

part (ii),

Y such that,

so that {/(/.)}

contradicts

4 \x^,

Let

• • • , x }.

fiV) = f)"*}\jc„ V.], ' • "z = l V i U £ 9> and g 6 ^. Since an element

\x .\ isa of

z£ ù)- Then by choice the continuity

of c/>.

f, g){ = 0, so that C (X, Y) must be of first category by the

Proposition. Part

(iii)

follows

"xeX Y , which

and using first

is dense

the fact

category.

from

considering

since

that a dense

G (X, Y) to be a subspace

X is completely

subspace

Hausdorff

of a space

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of first

of

with respect

category

to

is of

Y,

434

R. A. McCOY Corollary

1. Let

a convergent nondegenerate

first

X be a completely

sequence

which

pathwise

Hausdorff

is infinite

connected

Hausdorff

which

contains

of X, and let

space.

Y be a

Then

C (X, Y) is of

connected

completely

category.

Corollary

2. // X is a nondegenerate

pathwise

Hausdorff

space,

and

Y is a nondegenerate

Hausdorff

space,

then

C AX,

The first

space

corollary

is completely

follows

and the second

corollary Y are open

of

the union

Y) is of first

locally

with respect

follows

that

from the first

of open

connected

a completely

to a pathwise

category

subspaces

Hausdorff

connected

and the fact

and from the Banach

of any family

pathwise

category.

from the fact

Hausdorff

components that

space

as a subset

that

the path-

theorem,

of first

space,

category

which

says

is of first

category. The three

concepts

given

in statements

(i), (ii),

and (iii)

overlap,

but no two of them contain

the third.

Note

compact

space

(ii),

and every

quasiregular

(ii).

A closed

not a Baire

space

fails

space

satisfying

to satisfy does

(i) but not (ii) or (iii).

(ii) but not (i) or (iii), usual

open

countable topology

satisfy

sets

take

satisfies

o along

Finally,

interval

with

a countably

with sets

of the Theorem

every

sequentially

space

is

of a

of a space

satisfying

topology

generated

containing

infinite

which

is an example

For an example

X to be the reals

not containing

complements.

that

by the

o and having

set with the cofinite

(iii) but not (i) or (ii).

REFERENCES 1. M. K. Fort,

(1951), 34-35.

Jr.,

A note

on pointwise

convergence,

Proc.

Amer.

Math.

Soc.

Amer.

Math.

2

MR 12, 728.

2. R. C. Haworth and R. A. McCoy, Baire spaces (to appear). 3. M. R. Krom, Cartesian products of metric Baire spaces, Proc.

Soc. 42 (1973), 588-594. 4. R. A. McCoy, Function spaces with intervals as domain spaces, Fund. Math, (to appear). 5. —-, Baire spaces and hyperspaces, Pacific J. Math, (to appear). 6. J. C. Oxtoby, The Banach-Mazur game and Banach category theorem, Contributions to the Theory of Games, vol. 3, Ann. of Math. Studies, no. 39, Princeton

Univ. Press, Princeton, N.J., 1957, pp. 159-163. 7.

-,

Measure

and

category,

Springer-Verlag,

MR 20 #264. New

York,

1971.

DEPARTMENT OF MATHEMATICS, VIRGINIA POLYTECHNIC INSTITUTE AND STATE UNIVERSITY, BLACKSBURG, VIRGINIA 24061

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