Properties of Hyperconnected Spaces, Their Mappings Into Hausdorff Spaces and Embeddings Into Hyperconnected Spaces

Acta Math. Hung. 60 (1-2) (1992), 41-49. PROPERTIES OF HYPERCONNECTED SPACES, THEIR MAPPINGS INTO HAUSDORFF SPACES AND EMBEDDINGS INTO HYPERCONNECTED SPACES N. AJMAL and J. K. KOHLI (Delhi) 1. Introduction Professor Levine calls a space X a D-space [5] if every nonempty open subset of X is dense in X, or equivalently every pair of nonempty open sets in X intersect. In the literature D-spaces are frequently referred to as hyperconnected spaces (see for example [9], [10]). In this paper we extend the concept of hyperconnectedness to pointwise hyperconnectedness and use it to study the properties of hyperconnected spaces. We shall call a space X pointwise hyperconnected at z in X if each open set containing z is dense in X. It is immediate that a space X is hyperconnected if and only if it is pointwise hyperconnected at each of its points. It is clear from the definition that the property of being a hyperconnected space is open hereditary. In fact every subset of a hyperconnected space having a nonempty interior is hyperconnected in its relative topology. In particular, every/~-subset [6] of a hyperconnected space is hyperconnected. However, in the sequel, Example 2.3 shows that hyperconnectedness is not even closed hereditary. This corrects an error in [5] where it is erroneously stated that hyperconnectedness is hereditary (see [5, Theorem 2(1)]). More generally we shall show that every topological space can be realized as a closed subspace of a hyperconnected space (see Theorem 3.1). Section 2 is devoted to the properties of (pointwise) hyperconnected spaces. We show that hyperconnectedness is preserved under feebly continuous surjections and inversely preserved under feebly open injections while pointwise hyperconnectedness is invariant under continuous surjections. Moreover, we prove that (pointwise) hyperconnectedness is productive and that every subset of a hyperconnected space having a nonempty interior is hyperconnected. Furthermore, we show that a space is hyperconnected if and only if every feebly continuous function from it into a Hausdorff space is constant and that every continuous function from a pointwise hyperconnected space into a Hausdorff space is constant. In the process we improve/generalize certain results of Noiri [7], Pipitone and Russo [8], and Levine [5]. In Section 3, we show that every topological space can be realized as a closed subspace of a hyperconnected space called "hyperconnectification'. 42 N. AJMAL and J. K. KOHLI Beside discussing basic properties of hyperconnectifications, we also reflect upon their functorial nature. The closure of a subset A of a topological space X will be denoted either by A or ClxA or C1 A; and the interior of a subset B of X will be denoted either by B ° or intx B or int B. An open set in a space is said to be regular open if it is the interior of its closure. 2. Properties of (pointwise) hyperconnected spaces, characterizations and mappings into Hausdorff spaces First we give some illustrative examples which either reflect upon the theory or will be referred to in the sequel. EXAMPLE 2.1. Let X denote N, the set of natural numbers, endowed with the topology generated by taking basic neighbourhoods of each n E N the set {1, 2,... , n}. The space X is a To second countable, hyperconnected space. EXAMPLE 2.2. Let X -{a,b,c} and Z -- {{a},{b},{a,b},X,O}. The space (X, 2:) is pointwise hyperconnected at c but neither at a nor at b. EXAMPLE 2.3. Let Y denote the closed unit interval [0,1] equipped with the usual topology/~. Let X = Y U {w}, where w ~ Y. A topology on X is defined by declaring V C X to be open if either V is empty or V = UU (w}, for some U E /~. The space X is a hyperconnected space. The relative P topology of [0, ½[ which it inherits as a subspace of X coincides with the h J Euclidean topology. Thus the closed set [0, 1] is not hyperconnected in L J its relative topology. This example shows that hyperconnectedness is not a closed hereditary property. PROPOSITION 2.1. For a topological space X, the set of all points where X is pointwise hyperconnected is a closed subset of X. PROOF. Let F denote the set of all points where X is pointwise hyperconnected. To show that F is closed, we shall show that X - F is open. To this end, let x E X - F. Then there is an open set U containing x such that U ¢ X and so U C X-F. Thus X-Fbeingtheunion of open sets is open. THEOREM 2.2. For a topological space X the following statements are equivalent: (1) X is pointwise hyperconnected at x. (2) Every nonempty open set intersects every open set containing x. (3) Every open set containing x is connected. (4) Every closed subset of X not containing x is nowhere dense in X. Acta Matheraatica It~ngarica 60, 199~ .

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