4 research outputs found
Topological Scott Convergence Theorem
Recently, J. D. Lawson encouraged the domain theory community to consider the
scientific program of developing domain theory in the wider context of
spaces instead of restricting to posets. In this paper, we respond to this
calling with an attempt to formulate a topological version of the Scott
Convergence Theorem, i.e., an order-theoretic characterisation of those posets
for which the Scott-convergence is topological. To do this, we
make use of the replacement principle to create topological
analogues of well-known domain-theoretic concepts, e.g.,
-continuous spaces correspond to continuous posets, as
-convergence corresponds to -convergence. In this
paper, we consider two novel topological concepts, namely, the
-stable spaces and the spaces, and as a result we
obtain some necessary (respectively, sufficient) conditions under which the
convergence structure is topological
-quasicontinuous spaces
In this paper, as a common generalization of -continuous spaces and
-quasicontinuous posets, we introduce the concepts of
-quasicontinuous spaces and -convergence of nets for
arbitrary topological spaces by the cuts. Some characterizations of
-quasicontinuity of spaces are given. The main results are: (1) a space
is -quasicontinuous if and only if its weakly irreducible topology is
hypercontinuous under inclusion order; (2) A space is
-quasicontinuous if and only if the -convergence in
is topological
Topological Scott Convergence Theorem
Recently, J. D. Lawson encouraged the domain theory community to consider the
scientific program of developing domain theory in the wider context of
spaces instead of restricting to posets. In this paper, we respond to this
calling with an attempt to formulate a topological version of the Scott
Convergence Theorem, i.e., an order-theoretic characterisation of those posets
for which the Scott-convergence is topological. To do this, we
make use of the replacement principle to create topological
analogues of well-known domain-theoretic concepts, e.g.,
-continuous spaces correspond to continuous posets, as
-convergence corresponds to -convergence. In this
paper, we consider two novel topological concepts, namely, the
-stable spaces and the spaces, and as a result we
obtain some necessary (respectively, sufficient) conditions under which the
convergence structure is topological