ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE X X (1978) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO
Séria I: PRACE MATEMATYCZNE X X (1978)
E . P q l ( W a r s z a w a )
A non-paracompaet space whose countable product is perfectly normal
The aim of this note is to show th a t the countable product of the non-paracompact space E constructed by the author in [3], Example, is perfectly normal. This yields the following
E x a m p l e . A non-paracompact, first countable space E whose countable product E*0 is perfectly normal.
Moreover, E is locally metrizable and it is of cardinality Kl} and the product E H° is collectionwise normal.
Since this note supplements the author’s paper [3] and the reasonings we shall use are in essence natural modifications of ideas used in [3], we shall omit the details.
Our terminology and notation follows [1] and [3].
1. Let us recall the definition of E. Let W (сог) be the set of all count
able ordinals with the order topology. For every limit ordinal £ < а)г choose a sequence Ж->{а: а < £} such th a t sup{a?f(%): n e N} = £ and let E — {xs: £ < cox}. Let q be “the first difference” metric on the set E, i.e., for £ Ф rj we put
Q{æè i x n) min{w: xs(n) Ф xn{n)}’
and let я: E-> W (cq) assign to the ordinal £.
The topology of E is the maximum of the metric topology induced by q and the weakest topology making к continuous; or else, denoting by Eq the topological space with underlying set E and the topology in
duced by q, we have E = {{æè, £): £ < cox} <= E e xW ((o1).
2. By a theorem of K atëtov [1], Problem 2.7.15, we have to show th at for every n the product E n is perfectly normal. Let us observe th a t for this purpose it is enough to prove th a t for every n the space
T = {(a?0, *(®0) ^ i < n } œ E n+1
436 E. Pol
is perfectly normal. Indeed, T is a closed set (as и is continuous) and T is a GVset (as T = f | П U {(^0, son): oo0{j) V ook{i)} and hence T
k ^ n г j
is a 6Vset in the space E*+1), and moreoyer, E n+1 = U \ { T ) , where
<<»
h{: E n+1 -+En+1 is a homeomorphism given by the formula (a?0, . oc{, ...
..., (Л/г-, ... j so0, .. , f ocn).
3. We proceed by induction. Let us assume th a t the space E n is perfectly normal (for n = 1 this was actualy proved in [3]); as was ob
served in Section 2 we have to verify th a t the space T c E n+1 is perfectly normal.
Simultaneously with the topology of T we shall consider the weaker topology of the subspace of the product E e x E n, which is perfectly normal by the inductive assumption and a theorem of Morita [1], Problem 4.5.10;
we shall denote the space with underlying set T and this weaker topology by T*, and A* will stand for the closure of a set A a T in the space T*.
As in [3] the key point of our proof is the following lemma, which gives information about the difference
B (A ) = A * \ l .
Let л: T ^ E be the projection onto the first axis and let xn = но л:
T->W{o)x).
Le m m a (cf. [3], Lemma 2). For every set A a T the set x0[R(A)) is not stationary.
P ro o f. Assume on the contrary th a t the set 27' = x0{B(Aj) is sta
tionary. For every £ g27' let us choose a point Pe = ^(f)* •••»»/„({)) e F (A ).
Since P | g T we have £ > / < ( £ ) > for i < n. Using successively a Xeumer’s theorem on regressive functions on stationary sets (see [2], Appendix .1) one can choose a stationary set 27 <= 27', a set L c ( 1, ..., n} and, for every i e L, a point x y.gE such th a t
(£ if i ф L , (*) if £e27, then /<(£) =
(у г- it г e L .
Since gA* we can choose for every £ g27 a sequence {qf ) cz A which converges to p ? in the space T* ; let o f = л ((fit). The reasonings given in the proof of [3], Lemma 2, show th a t there exist Я g27 and a sequence (Ял) <=■ 27 with Xk/ Я such th a t g(a?A , жд)->0 and x(a™ ) < Я for m, k e N . Since, by (*), for every £ g27 and i < n the г-th coordinate of the point Pi is equal to a?{, whenever г $ L, and it is equal to ^ . — otherwise, we infer th a t the sequence (px ) converges to р л in the space T*. Let us choose a sequence (mk) cz N such th a t the sequence {qf£) converges to the point
Non-paracompact space 437
p k in the space T*. I t is easy to observe th a t ($£*) tends to p k also in the space T, as < A. We obtain th a t p k e E ( l ) n i = 0 , a contradiction.
4. We are ready now for the proof th a t T is perfectly normal (cf. [3], Proposition 1).
Let TP be an open subset of T. Let Г be a closed and cofinal set in TP(cox) disjoint from x0( B ( T \ TP)) (see Section 3) and let us put F = щ 1(Г), G = T \ F . Thus T \ W * n ( W n F ) — 0 and hence the open in T* set U = T \ ( T \ W ) * contains the set Wr\F. Since T* is perfectly normal we have U — [J Un, where Un are open in T* and U* cz U a W. Moreover,
П
the reasonings given in [3], the proof of Lemma 3, (11), show th a t the space G has a base c-discrete in T and therefore one can write TPnU
— U Vn, where Vn are open and Vn c TP. Finally, taking TPn = Un и Vn, П
we have TP = U Wn and Wn a TP, with TPn open. This proves th a t T П
is perfectly normal by a well-known lemma (see [1], Lemma 1.5).
5. The space E*° is in fact collectionwise normal. We sketch the proof of this assertion. By the remark in Section 2 we have to show th a t for each n e N the space T cz E n+l is collectionwise normal and, pro
ceeding by induction, we may assume th a t the space T* c E e x E n is collectionwise normal.
Let {Fa : s e 8} be a closed and discrete family of subsets of T. Using the arguments similar to those of [3], Bemark 2, one can verify th a t the set 2J = {*:0(IS(-FJ) : s e 8) is not stationary. As in Section 4, we choose a closed set F (F = «y1 (U), where U is a closed, cofinal set disjoint from Z) such th a t the space G = T \ F has a base or-discrete in T and B (F S) cz G for s e S. The sets F s n F are closed in the space F regarded as the subspace of T*. Using this fact, the collectionwise normality of T* and the property of G one can find open sets Wsn such th a t F 3 c: {J TPsn,
_ П
Wsnn F t = 0 if s Ф t, and each family {TP8ft: s e $} is discrete in T.
Finally, we put Gsn = TPan\ ( J Wtm' < Ф », m < n} and Gs = U 08n. The n
sets Ga are pairwise disjoint and F a cz Ga.
R e f e r e n c e s
[1] R. E n g e lk in g , General topology, Warszawa 1976.
[2] I. J u h â s z , Cardinal functions in topology, Amsterdam 1971.
[3] R. P o l, A perfectly normal, locally metrizable non-paracompact space, Fund.
Math. 97 (1977), p. 37-42.
