ON PERVIN ’ S EXAMPLE CONCERNING THE CONNECTED-OPEN TOPOLOGY

Irudayanathan and Naimpally [1] introduced a topology for function spaces (called the “connected-open” topology) which has the property that the connected functions form a closed set provided that the codomain is completely normal. Pervin [2] gave an example showing that the proviso cannot be weakened to normality. The purpose of this note is to point out a lacuna in his demonstration, and to re-establish the validity of the example.


INTRODUCTION.
Let X and Y denote topological spaces, and F the set of all mappings from X to Y.For each connected subset K of X and each pair U, V of open subsets of Y denote by W(K; U, V) the subset {f F f(K) U u V, f(K) n U # # # f(K) n V} of F. The collection S of all these sets W(K; U, V) is a subbase for the connected- open topology T on F, introduced by Irudayanathan and Naimpally in [I] where it is To show that normality of Y is not sufficient for this result, Pervin [2] took Y as a modification of the Tychonoff plank, with an open interval of reals interpolated between each ordinal and its successor in the construction; appealed to cardinality to obtain a function f from the unit interval X [O,I] onto a subset A* u B* of Y, where A* and B* were separated but had no disjoint neighbourhoods in Y, and where f-l({y}) was dense in X for every y in A* u B*; and proved that any member W(K; U, V) of S which contained f must also contain a connected function.However, this does not suffice to establish that the (non-connected) function f belongs to of a point to intersect a set without every basic neighbourhood doing so.We shall show that f is, nevertheless, a limit of connected (indeed, of continuous) functions. 2.

PERVIN'S EXAMPLE REVISITED.
Let J denote the connected, compact T 2 space formed from the second uncountable ordinal WD by interpolating a copy of (O,I) between each element (other than the maximum) and its successor, and imposing the order topology on the resulting chain; (Pervin's definition of these sets is incompatible with his assertion that they are connected; the above is presumably what was intended.)Considerations of cardinality establish the existence of a mapping f from ,I] onto A* u B* such that the preimage of each singleton is dense.It will now be shown that every neighbourhood of f contains a connected function.
Consider a typical basic T-neighbourhood G n{W(Ki; U i, V i) i--I, 2, n} of f where (for each i) K. is a connected subset of [O,i], U. and V. are open in Y, I I i and V. and meets them both.No loss of and f(K i) is contained in the union of U I 1 generality will be incurred by assuming that the sets K. are distinct since W(K; U, V) n W(K; U', V') W(K; U o U', V V').
Denoting by the number of degenerate intervals amongst the Ki, where O s n, we can arrange the labelling so that K. is a singleton for i and is non-degenerate i for i > j.The strategy of the proof is to determine a subset Z of Y of the form suggested by a876e8 in the diagram below (which see), where x is chosen to ensure that Z is contained in U. u V. for all i j, and z is selected so that Z includes I n V. for each i; a path-connectedness argument within Z will at least one point of U then produce a continuous function belonging to G. u V.. Thus for For i > j, f(K i) is the whole of A* u B* and is contained in U each positive integer n the product of compact sets {b}x[O,l-2-n] is contained in U. u V., and a lemma of A.D. Wallace (see [3], p.142) allows us to  and so if x denotes the maximum of the elements x. here chosen, we have for all i j.
(2.1) (In the event that n, i.e. that all the K. are degenerate, (2.1) may be obtained by an arbitrary choice of x b.) Still considering the case i j, we see from (2.1) that the connected set A* u (x,b]x[O,l) is contained in the union of U. and V. and intersects them both; 1 1 so it must be possible to choose a point t(i) (t(i) I, t(i) 2) of U i n V i such that either t(i) e A*, or else t(i) e (x,b]x[O,l): and in the latter case, the observations that U. n V. is a neighbourhood of t(i) and that b is not isolated in J I 1 will allow us to assume that x t(i)l < b.Turning now to the case i j, f(K i) is here a single point of (A* u B*) n U i s Vi; if this point lies in A* we denote it and consider the product space Y Jx[O,l].(The space W used here by Pervin instead of [O,I] is homeomorphic to [O,I] .)Denote by a and b (respectively) the least and greatest elements of J, and by A* and B* the following subsets of Y: A* [a,b){l}, B* {b}x [0, I).
[a,b) such that l,n b]x[O I-2 -n] cU.uV.. ,b) inherits from its cofinal subset Wfl \ {b} the property that each countable subset is bounded above: choosing then a strict upper bound x. < b for the sequence (xi, n) we see that [xi,b][0,1) !Ui u Vi;