GENERALIZED TOTALLY DISCONNECTEDNESS

In this paper totally disconnectedness is generalized to maximal disconnectedness, which is investigated, 
and additional properties of totally disconnectedncss and 0-dimensional are given.


INTRODUCTION
Totally dmconncctcd spaces were considered as early as 1921 by Knaster and Kuratowski  [1] and by Sierpinski [2].A space (X,T) is totally dmconnected iff the components of (X,T) are the points.Questions about, possible gencralzatmns of totally disconnected spaces led to the following discovery.TIIEOREM 1.Let (X, T) be a totally disconnected space.Then (X, T) is T.
PROOF: Since components are closed, then the singleton sets are closed, which implies (X, T) is T1.
Thus for Ta spaces, totally disconnectedness is maximal disconnectedness in the sense that the components are the smallest possible sets, which motivated the introduction and investigation of maximal disconnected spaces in this paper.
2. MAXIMAL DISCONNECTEDNESS.DEFINITION 1.Let (X, T) be a space and let C be a component of (X, T).Then C is a minimal componont of (X, T) iff C does not contain a nonempty proper closed connected subset.The space (X, T) is maximal disconnected iff the components of (X, T) are minimal components.
Note that for a space (X,T), Cl({z}) is connected for each z X and that a component C of (X,T) is a minimal component of (X,T) iff C Cl({x}) for each x X.
In 1961 A. Davis [3] was interested in properties R,_ weaker than T,, which together with T,_, would be equivalent to T, 1, 2. In the 1961 investigation R0 and R1 spaces were defined.A space (X, T) is R0 iffone of the following equivalent conditions is satisfied: (a) if O T and x O, then Cl({x}) C O, and (b) {Cl({x}) Ix X} is a decomposition of X.A space (X,T) is R iff for x,y X such that Cl({x}) Cl({y}), there exist disjoint open sets V and Y such that Cl({x}) C V and Cl({y}) C V. The 1961 paper [3] was a continuation of work done by N. Shanin in 1943 [4], in which R0 spaces were called weak regular spaces.Combining this information with the note above in a straightforward proof, which is omitted, gives the following result.
THEOREM 2. Let (X,T)be a space.Then (X, T)is maximal disconnected iff (X, T)is R0 and the components of (X, T) are closures of singleton sets.
The results above can be combined to obtain the following result.COROLLARY 3. Let (X, T) be a space.Then the following arc equivalent: (a) (X, T) is totally disconnected, C. DORSETT (b) (X, T) T max,real d,sconnecled, and (c) (X, T) is To max,real d,sconnocted TIIEOREM 4. l,et (X, 7') be space Then (X, T) s maxtmal dtsconnoctcd tff every honwonaorphtc mage of Y. 7") ,s nax,mal dsconnoclod.
Combining the results above with the fact that for a spe (X,T), (x(ro),Q(rO)) ro giv the following result.
COROLLARY 7. Let (X,T) be a space.Then (X,T) is maximal disconnected iff (X(TO), Q(TO)) is maximal disconnected.
l,et xs C' For each a A,o# :',let(-'a bcacomponentof(.a,Ta)andletxaG Ca. Then Ha,C'aisaclsed (' C/({}) Tiros (X,, E) is maxima1 disconnected Combining tle results above wh he fact that, the produc space of a nonempty collection of nonempty spaces s T ff each factor space s T gves the next result.
COOI, I,AY 11 l,et {(X,T) a A} be a nonempty collection of nonempty spaces.Then (X.T) otally disconnected for each a A iff ( X, W) is totally disconnected.
In S. Wllard's 1970 book [6] relationships between totally dsconnected and 0-dimensional were examined.A space (X, T) s 0-dimensional ff each point of X h a neighborhood be consmtmg of closed open sets.Below, results m Wllard's book are used not only to further investigate maxmal disconnectedness, but also, to flrther investigate totally dmconnectedness.
Snce every 0-dimensional space s completely regular, then every 0-dimensional space is regular, and thus TEOEM 2. Le (X,T) he a space.Then (X,T)is 0-aimens,ona iff (X(TO),O(TO))is 0-aimens,ona.PROOF: Suppose (X,T) s 0-dimensional.Let C X(TO).Le a C. Let O be a neighborhood be of coni,n of oea opon e. S,n P(TO) i oa, opt., .aonin.o.[71, be of C cons,st,ng of closed open sets.Thus (X(TO), Q(TO)) is 0-dimensional.Conversely, suppose (X(TO), O(TO)) is 0-dimensional.Let X.Let C X(TO) such that C. Let O be a neighborhood be of C consisting of closed open sets.Since P(TO)-(P(TO)(O)) O for each O T [7], P(TO)-(O) is a neighborhood be of consisting of closed open sets.Thus (X, T) is 0-dimensional.THEOREM 13.Let (X,T) be a 0-dimensional space.Then (X,T)is maximal disconnected.PnOOF.Since (X,T) is 0-dimensional, hen (X,) is n0 and (X(70),O(TO)) is 0-dimensional T, which implies (X(TO), Q(TO)) is otally disconnected [6] and (X, T) is maximal COrOLLArY 14.Every 0-dimensional T0 spe is totally disconnected.DEFINITION 2. A space is rim-compt iff eh of its points h a be of neighborhoods with compt frontier [6].
TIIEOREM 15.Let (X,T)be a space.Then (X,T)is rim-compact iff (X(TO),Q(TO))is rim-compact.PROOF: Suppose (X, T) is rim-compact.Let C X(TO).Let x C. Let O be a neighborhood base of x con- sisting of neighborhoods with compact frontiers.Let O O. Then Fr(Int(O)) is a closed subset of the compact set Fr(O), which implies Fr(Int(O)) is compact.Then P(TO)(Int(O)) Q(TO) and P(TO)-I(Fr(P(TO)(Int(O)))) Fr(P(TO)-I(P(TO)(Int(O)))) [7] Fr(Int(O))is compact, which implies Fr(P(TO)(Int(O)))is compact [7].
The lst result in thin seclaon follows mmediatcly from the fact that for T spaces, mctrzability and pseu- dometrzablty are equivalent COROLLARY 18.Let (X, T) be toally dmconnccted.Then (X, T) is mctrizable iff (X, T) is pseudometrizable.
Thus in results known for totally dmconneced metric spaces, metric can be replaced by pseudometric.