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## C OMPOSITIO M ATHEMATICA

o

### 2 (1975), p. 137-144

<http://www.numdam.org/item?id=CM_1975__30_2_137_0>

© Foundation Compositio Mathematica, 1975, tous droits réservés.

L’accès aux archives de la revue « Compositio Mathematica » (http:

//http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

http://www.numdam.org/

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137

REAL-VALUED FUNCTIONS ON CERTAIN SEMI-METRIC SPACES

Harold R. Bennett Noordhoff International Publishing

Printed in the Netherlands

H.

showed that if

### f

is a real-valued function on

Euclidean n-space

then

contains a dense

Y

on

such

### that f

restricted to Y is continuous. In this paper it is shown that

### if f

is a real-valued function on a

### regular

semi-metrizable Baire space

### X,

then X has a dense

Y such that

### J’

restricted to Y is continuous. Other

### questions

and extensions of

theorem are

in

and

In

the indicated

### result,

the concepts of First

sets and

Second

### Category

sets are crucial. The

theorem

in

page

is

used: If

is a

### family

of sets open relative to the union S =

and if each

is of the First

### Category,

then S is also of the First

### Category.

All undefined terms and notations are as in

1. Preliminaries

In the

definitions

### let , f ’ be

a real-valued function on a topo-

and let x e X.

D’EFINITION

The function

is said to

x First

### Categori- cally (written f1 ~ x)

if there is an e &#x3E; 0 and a

of

x such that

=

is a First

set in X.

DEFINITION

The function

said to

x Second

if

### given e

&#x3E; 0 then there exists a

of x such that

is a Second

### Category

set in X. The function

is said to

x Second

via R

via

if

&#x3E;

there is a

such that

n R is a Second

set in X.

(3)

138

DEFINITION

### (1.3):

An open set U is a

of a

### point

x if either x is in U or x is a limit

### point

of U.

It follows from Definition 1.2 that

if

### given

03B5 &#x3E; 0 there is a

### partial

U of x such that for any open subset V of

E V :

is a Second

subset of U.

DEFINITION

A function

is said to

x

x

if

### given 8

&#x3E; 0 there is a

of x such

that

=

is dense in

If x is a

limit

of

then

is said to

x

via R

x

via

if

n R is dense in

n R.

The

### following

is a useful characterization of Definition 1.4.

THEOREM

Let

be a

on a

space X.

then

x

and

each

U

limit

PROOF :

x

and U is any

### partial neighborhood

of x.

Let e &#x3E; 0 be

then x has a

such that

is dense in

### M(x, e).

Thus there exists z ~

n U such that

Hence

is a limit

of

### f ( U).

To show the converse, suppose

does not

x

### densely.

Then

there is an B &#x3E; 0 such that for each

N of x, the set

### {z~N:|f(x)-f(z) 03B5}

is not dense in N.

### Thus,

there is a non-empty open set

### UN

contained in N such that for all

8-

Then U =

N a

of

is a

of

x such that

is not a limit

of

### f ( U).

Let Z+ denote the set of natural numbers.

THEOREM

Let

be a real-valued

on a

space X.

=

X :

=

X :

does not

are sets

the First

in X.

PROOF: If x E

then there is an

&#x3E; 0 and a

of x such that

is a First

set. There is no

### generality

lost if it is assumed that

=

for some

### m(x)

E Z + . For each k E Z + let

=

=

and let

i E

be a countable

(4)

dense subset of

Let D =

If

let

and if x E

let

define

and if x e

### L(m, i),

let

If x and y are in R(m,

then

and

Thus

and are First

sets. Since

it follows that

a First

### Category

set. The theorem mentioned in the introduction was used in the

of this theorem.

### If x~F2,

then there exists

### e(x)

= e &#x3E; 0 such that for each

of x,

is not dense in

Let

### {r1, r2,···}

be the set of

rational numbers and

ri, let

It follows that

### F(i, j)

is nowhere dense for suppose 0 is an open set such that

~ 0. If p

are in

n

then

(5)

140

E

### 0 : f (p)- f(q)| 03B5(p)}

is dense in 0. This contradiction shows that

### F(i, j)

is nowhere dense and it follows that

is a First

### Category

set.

2. Semi-metrizable Baire spaces In the

### following

let all spaces be

spaces.

DEFINITION

A

space is a Baire

### Space

if the countable

intersection of open dense sets is a dense set.

THEOREM

### (2.2) : If

X is a Baire space and

is a real-valued

on

### X,

then there is a dense set D

on

such that

x E D, then

x

via D.

PROOF: Let

Theorem

is a First

set. Let

If

then

via

Let

E

does not

x

via

Theorem

is a First

set. Thus D =

1 ~

### F2)

is a residual set

### and,

since X is a Baire space, D is dense in X. If

x via

then

### f2 ~

x via D and if x E D then

x

via

### R1.

Let x E D. If e &#x3E; 0 is

and U n D is any

of x in D

is a

of x in

since

x

via

there exists

such that

Since U is a

of

q,

### {z~ U : |(z)-f(q)| e/21

n D is a nonempty Second

set.

### Let y

be any one of its elements. Then

Thus

is a limit

of

Theorem

via D.

DEFINITION

A

### topological

space is a semi-metric space if there is a function d with domain X x X and range a subset of the

### non-negative

real numbers such that

(6)

0 if and

if x = y, and

x is a limit

### point

of a set M if and

if

In

=

R. W. Heath has

shown the

### following equivalent

condition for a space to be a semi-metric space.

THEOREM

Let X be a

space and G =

n E

x E

a collection

open subset

X.

G

each x E

m E

is a

### non-increasing

local base at x, and

y E X

each n E

y

then the

### point

sequence x1, X2, ... converges to y.

Then X is a semi-metric space.

This theorem is a useful tool in the

theorem.

THEOREM

is a real valued

on a

### regular

semi-

metrizable Baire space

### X,

then there is a dense subset Y

X such that

### f

restricted to Y is continuous.

PROOF : Since X is semi-metrizable there exists a collection

of open subsets of X

and

### (ii)

of Theorem 2.5. Let D be a dense set in X such that if

then

x

### densely

via D. The

existence of D is

### guaranteed by

Theorem 2.2. Construct a discrete subset

of X and a

subcollection

=

a E

of G such that

for each

### 03B1~A(1), g(1,03B1), x(l, 03B1))

contains a dense subset

D

such that if

1.

To obtain

and

be a well

### ordering

of D and let e = 1 be

Let

### x(l, 1)

be the first element

and let

### n(1, 1)

be the first element of Z+ such that

is dense in

that

### x(l, fl)

has been chosen for each

(7)

142

03B4 such that

if oc

03B4 and

Let

### x(l, ô)

be the first element

such that

Let

### n(1, ô)

be the first element of Z + such that 1

and

is dense in

Let

### A(1)

be the set of all a which have been chosen in the process described above. Let

=

and let

Let

=

E

### A(1)}.

It follows that if x~

then

x

via

For if

### x~H(1),

then there exists a E

such that

### x~h(1, oc). Thus |f(x) - f(x(1, (X»1

= 1-03B4 for some b &#x3E; 0. But if x~

then x E D. Thus

b &#x3E;

there is a

of x such

that

is dense in

If

then

Thus z E

### Suppose B(l), - - -, B(k), G(1), ··· G(k), H(l), - - ., H(k)

have been chosen such that for 1 ~ i ~ k

~ ... ~

if g E

### G(i),

then g is a member of the local base for some element of

if g E

+

then there is a

E

such that

the elements of

are

(8)

D ~

=

where

and

is a

dense subset of

such that if

then

and

if x ~

then

x

via

To obtain

and

let

Let

and let

### n(k + 1, a)

be the first element of Z+ such that

and

is dense in Select from

a discrete subset

=

E

### A(k + 1, 03B1)}

and select from G a

collection

such that

if g E

then

for each

+

+

contains a dense

subset

such that if

then

and

Then let

(9)

144

and

It

### clearly

follows that the induction

is satisfied.

Let Y =

and for each

let

=

### U {g~G(n)}.

It follows that K =

### ~{K(n) : n~Z+}

is dense since each

### K(n)

is an open dense subset of X. Notice that Y is a dense subset of K for if z E

for each

there is an

such that

### z E g(n(i, 03B1i), x(i, (Xi» and,

sine X is a semi-metric space, the

sequence

### x(l, 03B11), x(2, (X2), ...

converges to z. Thus Y is dense in X.

Let x E Y and let e &#x3E; 0 be

Since

### x E Y,

there exists i~Z+ such that

for

### each j ~

i and there exists k E Z+ such that

E. Let

Since

### x E B(m), g(n, x) E G(m)

for some n~Z+ and if

e. Thus

### f

restricted to Y is a

continuous function.

DEFINITION

### (2.6):

A semi-metric space X is said to be

### weakly complete provided

there is a distance function d such that the

### topology

of X is

invariant with respect to d and

i E

is a monotonic

sequence

closed sets such

for each

there exists

a

of a

which contains

then

### n {Mn : n~Z+}

is non-void.

Standard arguments show that a

### regular weakly complete

semi-metric space is a Baire space. Thus the

is established.

CUROLLARY

is a real-valued

in a

### regular, weakly complete

semi-metric space

### X,

then X has a dense subset Y such that

### f

restricted to Y is continuous.

REFERENCES

[1] H. BLUMBERG: New Properties of All Real Functions. Trans. Amer. Math. Soc. 24

(1922) 113-128.

[2] J. C. BRADFORD and C. GOFFMAN: Metric Spaces in which Blumberg’s Theorem Holds.

Proc. Amer. Math. Soc. 11 (1960) 667-670.

[3] R. W. HEATH: On Certain First Countable Spaces. Topology Seminar, Wisconsin, 1965, 103-115.

[4] J. L. KELLY: General Topology. Van Nostrand, New York, 1955.

[5] K. KURATOWSKI: Topology. Academic Press, New York, 1966.

[6] R. LEVY: A totally ordered Baire space for which Blumberg’s theorem fails. Proc.

Amer. Math. Soc. 41 (1973) 304.

[7] R. LEVY: Strongly non-Blumberg Spaces. Gen. Top. and its Appl. 4 (1974) 173-178.

(Oblatum 2-V-1972 &#x26; 15-XI-1974) Department of Mathematics Texas Tech University LUBBOCK, Texas 79409

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