Generalizations of Topological Semilattices

Connectedness in Generalized Topological Semilattices and Lattices

“…Proof. We can show as in the proof of Theorem 2.2 of [15] that (« + l) i> continuity implies « b continuity for n ^ 2, and that 2* continuity implies sesqui b continuity; that sesqui* continuity implies l b continuity follows from Proposition 15 and the equality/(G) = \J{X g~l (G b ): geG} (each G e 0 ) .…”

“…Suppose {E, ^, T} satisfies the six conditions above. Firstly, (ii) implies that T itself is T 2 , and therefore 7\; it follows by Theorem 2.8 of [15], together with (iii) and (iv), and the observation that upper sesqui v continuity implies upper sesquicontinuity, that {E, ^, T} is 7\-ordered. We now show that, given a, b, c and d in E, the sets F abc , T ac and K abcd are closed.…”

“…Proof. If {£, ^, T} is 2-continuous or 2* continuous then by Theorems 2.2 and 2.10 of [15] it is u.s.c, and by Proposition 12(i) it is welded; the result of (i) follows by Proposition 17.…”

“…We shall use the following notation and terminology from [15]. Given a downdirected partially-ordered topological system {E, < , T } , A^E and xeE, we let &\ ^\ JV r (x)> c\ l [A] and inf [A] denote (respectively) the families of T-open sets, x-closed sets and t-neighbourhoods of x, and the r-closure and t-interior of A (usually omitting the superscripted T), and define L{A) = {xeE: x ^ a for all a e A} and D(A) = {x e E : x ^ a for some a e A}, while M(A) and I (A) are defined dually.…”

“…Continuity conditions on the lower-bound operation, definable for arbitrary downdirected partially-ordered topological spaces, and specializing to continuity of the infimum operation when applied to a (lower) semilattice, were introduced and studied in [15]. The purpose of the present paper is to show how certain theorems relating to connected topological lattices can be generalized to systems satisfying some of these continuity conditions and/or their duals.…”