A NOTE ON MAXIMALLY RESOLVABLE SPACES

A.G. El'kin [1] poses the question as to whether any uncountable cardinal number can be the dispersion character of a Hausdorff maximally resolvable space.

1980 AMS SUBJECT CLASSIFICATION CODE.54E35, 54D05, 54G20.I. INTRODUCTION.In the sequel we denote by 0 the first infinite cardinal number and by the first uncountable cardinal number.
For a topological space X, the dispersion character A(X) of X is the least among the cardinals of nonvold opens sets.
The space X is called maximally resolvable (Ceder  [2]) if it has isolated points or X is the union of A(X)pairwise disjoint sets, called resolvants each of which intersects each nonvold open set in at least A(X) points.
A topological space X is said to be I) Urysohn, if for every two distinct points x,y of X there exist open neighborhoods V,U of the points x,y such that V N U , 2) Regular at a point x, if for every open neighborhood U of x, there exists an open neighborhood V of x such that V c_u. 3) Almost regular, if there exists a dense subset of X at every point of which the space X Is regular.
Let X be a set and let {yi: i e I} be a family of subsets of X with each having a topology.Assume that for every (i, J) e I I, both l) the topologies of YJ agree on YIOYJ and 2) each yl yJ is open in yl and in YJ.Then the weak topology in X induced by {yI: i e I} is T {U: UOY I is open in yi for every i I}.

Every cardinal number
) I can be the dispersion character of a metric connected, locally connected space.PROOF.
Let (X,d) be a metric connected locally connected space wlth dispersion caracter I" We first construct a sequence of sets X 0, XI,...Xn,..., then we define a metric d* on the set Y IJ X and we prove that (Y,d*) is the required space.
n For an arbitrary point x of X we set x 0 {x}.
X\{x} and we consider the set x {x) u u yi(x) where I is an index set such that II and yi(x) is the i-copy of Y(x) attached to the point x.Assume also that Yi(x)l N YJ(x) for every i,J e I I, i J.
$Imilarly the set X 2 is defined as where 12 is an index set such that I121 , Y(x) X\ {x} and yi(x) Is the i-copy of Y(x) attached t the point x of XI\XO.Assume also that Yi(x)oYJ(x) for every I,J I UI2, i J and that Yi(x) flYJ(y) for every x y, i,J

I IU 12
Using induction, the set X n is defined as U Yi(x)), nffi3,4 .... where I n is an index set such that llnl N Y(x) X\{x} and yi(x) is the i-copy of Y(x) attached to the point x of Xn_l\xn-2" (It should be observed that to every point x of Xn_!\Xn_ 2 are attached M palrwlse disjoint copies of Y(x)).Assume also that n Yi(x) NYJ(x)" for every I,j e U Ik, i J and that yi(y) N yj(y)" for n k=l every x # y, i,j e U I k- We consider the set Y U X on which we define a metric d* as follows: Let a,b n n=l be two arbitrary points of Y and n,m be the minimal integers for which a e Xn, b Xm.Suppose n < m and let Yi(an_I), i I n, an_le Xn_l\Xn_ 2 and yJ (bin_ I), j e Ira, bm_ e Xm_l\Xm_ 2 be the copies of Y(an_I) and Y(bm_I), where the points a,b belong respectively.

+ d( ). m-I bm-i'bm-(i+l)
If a b then continuing the above process in a "parallel" way for both points a,b n n we find a finite number of points a n-l' an-2' "''an-k and a finite number of points bn_l, bn_2...bn_k such that an_ k --bn_ k for some k, It is easily verified that d* is a metric for the set Y and that (Y,d*) is a connected locally connected space with dispersion character M. COROLLARY 2.1.Every cardinal number )N 0 can be the dispersion character of a Hausdorff (resp.Urysohn, almost regular) maximally resolvable space Y with the following properties I) Every continuous real-valued function of Y is constant.
2) For every point a of Y, every open neighborhood U of a contains an open neighborhood V of a such that every continuous real-valued function of V is constant. PROOF.
Let (X,) be a countable connected, locally connected Hausdorff (resp.Urysohn, almost regular) space (lliadis and Tzannes 4]).We construct the set Y as in Theorem 2. above and we consider the space (Y,z) where T is the weak topology induced by the (palrwlse disjoint) spaces yi(x) where x e Xn\Xn_ i e I llnln' n--l,2,...It can easily be proved that Y is Hausdorff (resp.Urysohn, or almost regular) having dispersion character M.
In order to prove that Y is maximally resolvable we fix an index I for every union is Y\{x}, hence Y\{x} is maximally resolvable and therefore Y is maximally resolvable.
We now prove that every continuous real-valued function f of Y is constant (and hence Y is connected).Let a be an arbitrary point of Y and n be the minimal integer for which a e X.
The point a belongs to a copy yi n (an_l), i e In attached to the point a of The pace yi n-I Xn_l\Xn_ 2 s (an_l)U {an_l} is homeomorphic to X and since X is countably connected it follows that f(yi(an_ 1) U {an_l} c.Similarly, the point a belongs to a copy YJ(an_2) j In_ attached to an_ 2 of Xn_2\Xn_ 3 and n-I f(YJ(an_2) U {an_2}) c.
It is obvious that the point an_(n_l belongs to yk(x), where k E I and {x} X 0 hence f(yk(x)U {x}) c and therefore f(a)=f(x) c, for every point a of Y.
Similarly is proved property (2) (and hence Y is also locally connected).

COROLLARY 2.2.
There exists a countable connected locally connected Hausdorff (resp.Urysohn or almost regular) maximally resolvable space (not satisfying the first axiom of countability).
PROOF.Let Z be a countable Hausdorff (or Urysohn space not satisfying the first axiom of countability.We first embed Z in a countable Hausdorff (or Urysohn) almost regular space X [4, Corollary I].Then, we construct the space Y as in Corollary 2.1 above considering II =n 0 for every n--l,2, Since Z c_ X c_y it follows that at every point x Z the space Y does not satisfy the first axiom of countability. If a,an_k)+d*(b,bn_k).
n n--l,2,..., and we consider the sets and Dla__ ya(x ), e I n 1,2,...,} and we observe that D consists of palrwise n disjoint dense sets each of which intersects every open set in points and whose