Some Remarks on Weak Continuity

Another note on Levine′s decomposition of continuity

“…But first we observe perhaps the most notable consequence of (.). The following decomposition of continuity improves the result of Chew and Tong [6] as well as Theorem 5 of [5]. f(x) f(x) for each x E X, and note that fa has (,) whenever f has (,).…”

“…and (**) are introduced each strictly weaker than basic relative continuity and with (**) strictly weaker than (. Weak c-continuity was introduced by Noiri [4] and studied further in [3], and basic relative continuity was introduced in [5] In [6] it is shown that weak continuity together with interiority implies continuity. The reader will immediately see that the condition of interiority is equivalent to the condition that for each open set PROOF.…”

“…The reader will immediately see that the condition of interiority is equivalent to the condition that for each open set PROOF. From [6], f is continuous so that f(X) is connected and discrete. Then each C_ f(X) is clopen (closed and open) in the subspace f(X), so that f(X) is a nonempty singleton set.…”

“…More recently, in [5] local w* continuity was replaced by the strictly weaker local relative continuity which in this paper is called basic relative continuity. It is interesting that relative continuity as introduced by Chew and Tong in [6] is equivalent to Levine's w" continuity. Its real merit however, lies in the fact that its basic (or local) version can replace local w* continuity in many applications while being strictly weaker.…”

Decomposition of openness

“…These conditions when combined with weak openness in one case and almost openness in the other imply openness. The conditions are analogous to the interiority condition defined by Chew and Tong in [2] …”