Vol. XXV, 1974 657

A Note on Countably Generated Spaces

By

V. KA~XAX

The following two definitions are due to g . C. Moore and S. G. Mrowka [3]:

Definition. A topological space X is said to be determined by countable sets if A c X is closed whenever A contains the closure of each of its countable sets.

Definition. A topolo~cal space X is said to be determined by countable closed sets, if A c X is closed, whenever A contains the closure of each of its subsets with count- able closure.

Following many other topologists, we use the term "c-space" to denote a to- pological space determined by countable sets. Moore and Mrowka [3] asked whether every Hausdorff c-space is determined by countable closed sets. Franklin [1] gave a negative answer, by constructing a counter-example. But the space constructed by him was not regular. He asked whether every regular c-space is determined by countable closed sets.

All the known examples of c-spaces not determined by countable closed sets (e. g., the c-co-reflection of fl N, etc.) seem to support an affirmative answer. Further, we can prove the following result, without much difficulty.

Proposition. Every locally countable regular space is determined by countable closed sets.

R e m a r k . I t is known (see e.g. [2]) that c-spaces are precisely the quotients of locally countable spaces. I t can be easily seen that the class of spaces determined by countable closed sets, lies in between the class of locally countable spaces and the class of c-spaces.

Hence the above proposition makes one expect that every regular c-space is determined by countable closed sets. But we give a counter-example to prove the contrary:

Example. For each n = 1, 2, . . . , let Rn be a copy of the real line R. Let the Rn's be pairwise disjoint. Let ~ be an extra point. Let

Then on X, we define a topology as follows:

Archly der Mathematik XXV 42

658 V. KANNAN ARCH. MATH.

Each Rn is declared to be open, retaining its (usual) topology. We have only to specify the neighbourhoods of c~. A subset A of X containing r will be an open neighbourhood of co, in our topology, if and only if it satisfies the following two conditions :

(i) Rn ~ A is open in Rn for each n---- 1, 2 , . . . . (it) /~n \ A h a s a finite Lebesgue measure for all but a finite number of values of n.

(We can transfer the Lebesgue measure of R to g n via a fixed homeomorphism.)

Proposition. The above space X is a T3 c-space, but it is not determined by countable closed sets.

P r o o f . I t is easily checked tha t X is a Hausdorff c-space. The regulari ty of X at points other than r is obvious. To verify at c~, let V be

an open neighbourhood of ~ . Then there exists an integer no such tha t ~ n \ V has finite Lebesgue measure for every n ~ no. By the outer regular i ty of Lebesgue measure and the normal i ty of the real line, it is possible to choose an open set Wn in /~n such tha t Wn ~ R n \ V and such tha t ~ n also has finite Lebesgue measure, for every n ~ no. Now let

.

Then W is an open neighbourhood of ~ . Fur ther it can be checked tha t l~ c V. This proves tha t X is a regular space.

Final ly we show tha t X is not determined by countable closed sets. I f B is any subset of X \ { ~ } with countable closure, then /~ (h Rn is countable and hence has measure zero, for each n : 1, 2, . . . . Therefore (X\/~) ~) {c~} is a neighbour- hood of c~ and therefore c~ is not in ~. Thus X \ { c ~ } contains the closure of each of its subsets having countable closure. But clearly X \ { r is no t closed.

R e m a r k . The space of our example is easily seen to be heredi tar i ly separable and hereditari ly Lindelhf. Therefore it is hereditari ly paraeompaet and completely normal. Thus we have T5 c-spaces tha t are not determined by countable closed sets. This is much more than tha t is required in [1].

References

[1] S. P. Ft~.~KLI~, On two questions of Moore and Mrowka. Proc. Amer. Math. Soc. 21, 597 --599 (1969).

[2] V. KAN~AN, Coreflexive subcategories in Topology. Ph. D. Thesis, Madurai University, 1970. [3] R. C. MOORE and S. G. MROWKA, Topologies determined by countable objects. ]Notices Amer.

Math. Soc. 11, 554, # 614--688 (1964).

Eingegangen am7.1.1974") Anschrift des Autors:

V. Kannan Department of Mathematics Madurai University Madurai -- 625021 Ind,.'a

*) Eine revidierte Fassung ging am 4. 4. 1974 ein.