THE AT WORST REMOVABLE AND DISCONTINUITIES CLUSTER OF FUNCTIONS SETS T. R. Hamlett (received 21 January 1974; revised 25 March 1974) Abstract. f :X + Y A function x removable discontinuity at X-{a;} Y where X in iff for every filterbase 8 -*■x , we have that such that is Hausdorff has an at worst /(B) converges. 8 on Sufficient conditions for a function with an at worst removable discontinuity at x x to actually be continuous at are investigated. The relationship between functions which have an at worst removable discontinuity at and a generalized notion of a cluster set of 1. X function from K If Introduction. into and X* f at we have that p^ X* , then / exists. is a f is defined in [3, Definition 3.2] of points different from lim/(p^) is also explored. are Hausdorff spaces and p as having an at worst removable discontinuity at every sequence x x p in X if for such that limp^ = p , In this paper we extend this notion by using filterbases rather than sequences. We also generalize the concept of cluster sets and show their relationship to functions which have an at worst removable discontinuity. Cl 04) denote the neighborhood system at 2. A will be used to denote the closure of A function is a connected subset of Let will x . f from a space X into a space said to be connected if for every connected subset Theorem 2.2. W(ar) We will use the following definition and theorems. Preliminaries. Definition 2.1. and f C of Y is X , f(C') Y . be a connected mapping of the space Math. Chronicle 4(1976), 94-100 94 X into the T^-space Y . If contained in Proof. C is a connected subset of X , then /(C1(C)) is Cl(,f(C)) • Observe that it is sufficient for Y to be T in the proof given in [3, Theorem 3.3], Let Theorem 2.3. x be c.ny point of the space compact subset of the space set U in Y . X and let K be a Suppose that corresponding to each W(x) , a closed set F(U ) in Y is prescribed in such a way that (i) F(U) c F(V) (ii) 0(^(1/) 3. U in W(x) K . such that F(U ) does not meet [5, Lemma, page 436]. At worst removable discontinuity. A function Definition 3.1. Y does not meet : £/€N(a:)} Then there is a set X U c v , whenever f of a space X into a Hausdorff space is said to have an at worst removable discontinuity (abbreviated a.w.r.d.) at a point such that 8 x in X if for every filterbase x , we have that converges to lim /(B) 8 X - {x} on exists. It is well known [l, Remark 1, page 213] that the concepts of convergence based on filterbases and on nets are equivalent. It frequently happens, in a given situation, that one of these two methods for expressing convergence is more convenient than the other. An equivalent formulation of Definition 3.1 in terms of nets reads as follows: A function / of a space an a.w.r.d. at x S : D -*■X - {a:} such that in X X into a Hausdorff space D if for every directed set S converges to Y has and net x , the net fS:D-+Y. converges. Let Theorem 3.2. space Y . point y Then in converging to Y f f be a mapping of the space has cm a.w.r.d. at x in such that for every filterbase x , we have that f(B) 95 X X into the Hausdorff iff there exists a 8 converges to on X - {x} y , with 8 f Proof. Assume that and are filterbases on A f (B) and Let Bfl A Bfl A x X . Suppose that in X - {ar} which converge to z converge to distinct points {B IM denote the filterbase X - {x} is a filterbase on f(B nA) ¥ /(A) has an a.w.r.d. at f(B HA) y and such that respectively. : B t B and A €A} . Clearly, x , Now which converges to z is a filterbase which accumulates to both is Hausdorff, x B y , Since and cannot converge and we have a contradiction. Since sufficiency is obvious, the proof is complete. Example 3.3. [0,ft] If we let f be the function from the ordinal space fix') = 1 (see [l]> page 66, Ex, 5) defined by /(ft) = 0 , f then is not continuous at if i / 2 at ft . Observe that no sequences of points distinct from to ft . Example 3.4. X Let X topology on and ft but does have a.w.r.d. be the real line and let T ft converge be the smallest generated by the standard Euclidean topology S X on together with the topology of countable complements C[4], Ex. 63). 0 Note that a set is countable. countable. f (x) = 0 is open in f : {X,T) x < 0 , f(x) = 1 if p . 0 - U-A iff X Also, observe that Now let real number T is Hausdorff and (X,S) if x > 0 , It is easily seen that f A is not first and at 0 is not ''removable". f( 0) = p for some vacuously satisfies the defined at 0 so that it will be continuous. f X and be the function defined by requirements for an a.w.r.d. at 0 given in [3], continuity of U Z S where but cannot be re In other words the dis Part 2 of the proof of the following theorem shows that this problem is eliminated by our defini tion. The following theorem is analogous to Theorem 3.6 of [3], Theorem 3.5. space X in iff f X Proof. Let f be a connected map from the locally connected into the Hausdorff space has an a.w.r.d. at Observe that we may assume Y . Then f is continuous at x x . 0 - {x} is nonempty for every neigh 0 borhood at of is continuous x . The proof is in two parts. Part 1. f Assume y f- f{x) We assume that U Let N(x) - {x} V and respectively. Let converges to neighborhood 0 f(C-{x}) c X - {x} y f(0 - {x}) c JJ , W(x) such that C and Now let A x . We may assume that A - {x} have that f(x~) €V f(A - {x}) c w . f{A - {x}) If converges to If In either case, we have that continuous at X - {x} /(£) then f( A) is a filterbase on /(A) converges to which converges to by Part 1. W If in A ~ {x} = A x €A , = fUA-{x}) U {x» rily, we conclude that which converges to A - {x} = {A - {x} : *4 €A} = f[A) c W . f(A) X be any filterbase on x £ A , 0 . Now we y = fix') . f(x) , then there exists an A neighborhood of be an open which contradicts the connectivity is a filterbase on /(A-{a;}) /(x) x . Hence C Let is contained in ; otherwise, we have immediately that Since and which implies that there exists an open f . Thus we conclude that Part 2. y which converges to such that jj x . \0 - {x} \ 0€\'(x)}, denote the filterbase x of be the unique X - {a;} which converge to be disjoint open neighborhoods of K'(x) - {x} /(N(x) - {x}) have y and proceed to a contradiction. is a filterbase on connected set in x . Let has an a.w.r.d. at limit of all images of filterbases on of f x . Otherwise we have immediately that A f(x) . x } we is a such that and we have then we have U {f (x)} c W . = /U-{x» f(A) c W . Since converges to W f(x ) was chosen arbitra and that f is x . Since necessity is obvious, the proof is complete. Remark. It is clear from Part 2 of the above proof that a function which has a.w.r.d. at a point re-defined at define f(x ) x x and is net continuous at so that it will be continuous at x . x can be Indeed, re to be the unique limit of all images of filterbases 97 B on X - {s;} 4. Cluster sets. such that B x . converges to The following definition is given in [2, Definition 3] for real valued functions. If Definition 4.1. Y , we say that y f X is a function from the space into the space is an element of the cluster set of f at x , C(f;x) , if a: is an element of the derived set of the denoted y . inverse image of every neighborhood of If Theorem 4.2. f is a function from X into the following are J , equivalent : (i) y (ii) There exists a filterbase (iii) is an element of B converges to y is an element of C(f',x) , x and on B f(B ) X - {x} such that converges to y . fl{Cl(/(£/ - {#})) : J/€W(s:)} . The proof is straightforward and hence omitted. Theorem 4.3. If the space into the compact space Let Proof. Since A X I B x is a limit point of be a filterbase on V Now if /"(B) J C(fyx) subordinate to is nonempty. which converges to f(B ) y in x . X . Let which converges to y . : U t N(x ) and 4 €A} , V satisfies condition (ii) of Theorem 4.2, and y €C(f;x) . Theorem 4.4. X Let f be a connected mapping of the locally connected into the T,-space neighborhood U of x is on element of Proof. is a function from denotes the filterbase it is easily seen that space f has an accumulation point {(tf-fo}) fl f 1 ^ ) hence and I , then X - {x} be a filterbase on is compact, X and J . x If is connected for every is a limit point of X , then f(x) C(f;x) . By Theorem 2.2 we have that Cl (f(U - { a : } ) ) U - {x} for every U in /(Cl (U - {x})) U(x) . 98 is contained in If we suppose that f(x) is V not in C(/;x) , then there exists a not in Cl(/(V- {x})) . This implies that /(Cl(7-{x})) and consequently 0 exists an in W(x) 0[\V = {x} , that x N(x) is not in x fix') such that /(x) is is not in Cl(7-{x}) . Thus there 0 H (7 - {x}) such that and hence in is empty. We conclude X is not a limit point of in contra diction to our assumption. Let Theorem 4.5. Hausdorff space a.w,r,d. at f be a mapping of the space Y , If x x iff C(f',x) f X is a limit point of into the compact X } then f is a singleton , Proof. Assume that C(f;x ) is nonempty and by Theorem 3,2 it must be a singleton, has an a.w.r.d. at X - {x} N(x) - {x} , V which converges to X , the collection point of X - {x} intersect. does not intersect conclude that /(N(x) - {a;}) Thus /(B) x If converges to f y and Proof. which U in Y does not such that f(U - {x}) c V . We y . Moreover, since is subordinate to / is compact C(/;x) M(x) x . Let 8 /(N(x) - {x}) . has an a.w.r.d,, at x . is a connected map from a locally connected space into a compact Hausdorff space f Y . Since Y Y - V . Hence X - {x} , /(B) on converges to Corollary 4.6. then in By Theorem 2.3, there exists a converges to be a 8 x , Since x is a limit which converges to is a compact set in Cl {f{U- {x})) X y be an arbitrary neighborhood of Y -V and let {u - {x} : £/€N(x)} , denoted by is a filterbase on and Hausdorff, x , By Theorem 4,3, C(f;x ) = {y} To show sufficiency, assume that filterbase on has an is continuous at x Necessity is obvious. iff Y and C(/;x) x is a limit point of is a singleton . If we assume that ton, then Theorem 4.5 implies that / C(/;x) has an a.w.r.d. at Theorem 3.5, this is sufficient to conclude that / is a single x . By is continuous at x . Acknowledgement. The author wishes to thank Paul E. Long for his patience and guidance. 99 X , REFERENCES 1. J. Dugundji, Topology , Allyn and Bacon, Boston, 1966. 2. Ulysses Hunter, An abstract formulation of some theorems on cluster sets , Proc. Amer. Math. Soc. 16(1965), 909-912. 3. William J. Pervin and Norman E. Levine, Connected mappings of Hausdorff spaces , Proc. Amer. Math. Soc. 9(1958), 488-496. 4. Lynn A. Steen and J. Arthur Seebach, Jr, Counterexamples in Topology, Holt, Rinehart and Winston, Inc., New York, 1970. 5. J. D. Weston, Some theorems on cluster sets, J. London Math. Soc. 33(1958), 435-441. University of Arkansas

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