TWO COUNTABLE HAUSDORFF ALMOST REGULAR SPACES EVERY CONTINUOUS MAP OF WHICH INTO EVERY URYSOHN SPACE IS CONSTANT

We construct two countable, Hausdorff, almost regular spaces 
I(S), I(T) having the following properties: (1) Every continuous map of 
I(S) (resp, I(T)) into every Urysohn space is constant (hence, both 
spaces are connected). (2) For every point of I(S) (resp. of I(T)) and 
for every open neighbourhood U of this point there exists an open 
neighbourhood V of it such that V⫅U and every continuous map of V into 
every Urysohn space is constant (hence both spaces are locally 
connected). (3) The space I(S) is first countable and the space I(T) nowhere first countable. A consequence of the above is the construction 
of two countable, (connected) Hausdorff, almost regular spaces with a 
dispersion point and similar properties. Unfortunately, none of these 
spaces is Urysohn.


I. INTRODUCTION. lliadis and Tzannes
[1] posed the question whether for every countable Hausdorff space R there exists a countable Hausdorff (Urysohn, almost regular) space I(X) having the following properties: (I) Every continuous map of I(X) into the given space R is constant (-) For every point x of X and for every open neighbourhood U of x there exists an open neigbourhood V of x such that V_U and every continuous map of V into the given space R is constant.
(Spaces having properties (I) and (P-) are called in [I], R-monolithic and locally R-monollthic, respectively, and by their construction are connected and locally connected).
It is obvious that the above mentioned spaces I(S),I(T) answer partially this question (in case the countable space R is Urysohn) because both have properties (I) and ( 2) for every Urysohn space.
For countable spaces, first countable or nowhere first countable, connected, locally connected Hausdorff or Urysohn, almost regular or V. TZANNES having a dispersion point, or with other properties see [1]-[97].
A space X is called I) Urysohn,if for every two distinct points x,y of X there exist open neighbourhoods V, U of the points x,y such that VNU=I 2.) Almost regular if it has a dense subset of regular points.
A point p of a space X is called a regular point for, that X is regular at p) if for every open neighbourhood U of p there exists an open neighbourhood V of p such that Vc_u.
A point p of a connected space X is called a dispersion point if the space X\{p} is totally disconnected.
Let X be a set and let {X iI} be a family of subsets of X with each X. having a topology.Assume that for every (i,j)IxI both i) The topologies of X, X, agree on X C(, 2.) Each X 6IX is open in X and in X Then the weak topology in X induced by {X :iI> is z={U:UCX is open in X for every iI}. 2..

AN AUXILIARY SPACE.
The following space X which is due to Urysohn [35], will be used for the construction of two auxiliary spaces S,T which with the help of the embedding described in [I] will yield the required spaces.

J=l
The countable space X has the following properties: (I) It is Hausdorff, almost regular (all points of X besides a,b are regular points).
( .)f(a)=f(b), for every continuous map f of X into every Urysohn space,f because the points a,b can not be separated by disjoint closed ne ighbourhoods ).
be disjoint copies of X and let a ,b be the copies of a,b, respectively in the space X For every n=1,2.
we attach the space X to the space X identifying the point b with a .le set a =x b =a =x n=l,2.f(p )=f(p+) for every continuous map f oi" S into every Urysohn space). 3.
There exists a countable, first countable Hausdorff, almost regular space I(S) having the following properties: (1) Every continuous map of I(S) into every Urysohn space is constant (hence I(S) is connected).
(2.) For every point s of I(S) and for every open neighbourhood U of s, there exists an open neighbourhood V of s, such that VcU and every continuous map of V into every Urysohn space is constant (hence I(S) is locally, connected).PROOF Let S be the space consrtucted above.
Ne set J=S\<p A =SxS(S) A(S)={(x,y)SxS:x=y} and we constructas in Ill,first the  That it is first countable follows by the fact that S is first countable (all spaces X,Y,21 are first countable) and by relation (7) [1,Theorem I] which in this case becomes X(I(S))=max{x(S), }= 0 0 In order to prove Properties (I) and Ca), observe that by the property of the space S (that f(p-)=f(p/), for every continuous map of S into every Urysohn space) and by the definition of topology on In(S,A it follows that I) Every continuous map of In(SA that V_cU and every continuous map of V into every Urysohn space, is constant on VNI"-i(S,A ).Finally, by the definition of topology on I(S) it follows that l) Every continuous map of I(S) into every Urysohn space is constant (hence I(S) is connected because the set of real-numbers with the usual topology is a Urysohn space) and 2.) For eve- ry point s of I(S) and for every open neighbourhood U of s there exists an open neighboorhood V of s such that V_U and every continuous map of V into every Urysohn space is constant (hence I(S) is locally connected).
REMARK B.I.If we consider as initial space the space X of Section I then the resulting space I(X) will be a countable, first countable, Hausdorff, anti-Urysohn space having Properties (I) and (2.), (a space S is called anti-Urysohn if for every x, yS and for every open neighbourhoods U,V of x,y respectively, UNVxO).In-1(8,A Is not first countable at every point of I"-*(8,A ). the space If S) will be nowhere first countable.It is easy to prove that while I(S) is Hausdorff almost regular having Property (I), it is not locally connected (hence does not have Property (2.)).PROPOSITION 3.2..There exists a countable, nowhere first countable, Hausdorff, almost regular space having Properties (I) and (2.) of Proposltion 3. I.

PROOF.
consider the space M=U{p}, pmkN where is the set of natural numbers and eq is the Stone-Cech compactification of .The space M is countable regular and not first countable at the point p.
be the copies Let M M be two disjoint copies of M and let P,'Pa of p in M,,Mz, respectively.Re attach the copies M,,Mz to the space S to pand the point Pa to p Re consider the attaching the point p, Obviously, the space T is Hausdorff almost regular not first countable at the points p p and f(p )=ffp ) for every continuous map f into every Urysohn space.
The space I(T) constructed as in Proposition 3.1 is the required space.
COROLLARY 3. I.There exists a countable, first countable, (or nowhere first countable) Hausdor f f, almost regular space having the following properties: (I) Every continuous map of it into every Urysohn space is constant (hence it is connected).
(P.)It has a dispersiou point.
( 3) For every open nelghbourhood U of the dispersion point there exists an open neighbourhood V of it such that every continuous map of V into every Urysohn space is constant (hence it is locally connected only at the dispersion point). PROOF.
First we observe that both spaces S and T of Section 2. and Proposition 3.3, respectively are tota 1 ly disconnected.Hence, we can apply [I, Theorem 2] using as initial space the s pace S, for the construction of countable first countable and the space T, for the construction of the countable nowhere first countable space.
The other properties of both spaces are proved as in Proposition 3. i.
.> [J _(X \<a ,b >) 0 we add one more point p.On the set Z=YU(p} we define the basis of open neighbourhoods of the point p to be the sets TWO COUNTABLE HAUSDORFF ALHOST REGULAR SPACES ?11 u (p)=(x-i>_n>U __(X "{a ,b ')U U<x n=1.2.
is Hausdorff almost regular is proved as in[l,Lemma 2.].
Urysohn space is constant on In-*(S,A and 2.) For every point n REMARK B.a.If on the set I*(S,A we define the topology to be 0 the weak topology induced by the spaces RA and S then the space o II(S,A is not first countable at ever7 point of S. Hence the set .) ffp )=f(p ), for every continuous ;nap f of S into every Urysohn space. (To prove this observe that since S is not Urysohn at every pair {x ,O,x :n=l,2.}areregular points).(