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Spread Uniformities and Uniform Spreads
Abstract
We discuss a point free analog of the topological fact that
every uniformly continuous function on a dense subspace of a uniform space
into a complete uniform space has a unique continuous extension. It is shown
that a uniform frame homomorphism h: L → M from
a complete uniform frame L into a uniform frame M (not necessarily
complete) onto which a dense
surjection ρ: N → M maps has a unique
uniform extension g : L→ N such that ρ ο g = h. We introduce spread uniformities
and uniform spreads into the theory of uniform frames, and, then, show that
the extension so constructed is a uniform spread.
Mathematics Subject Classification (1991): 54D15, 54D20, 54E15
Keywords: uniform frame, uniform local connectedness, spread uniformity,
uniform spread, spread omomorphism, higher separation axioms, noncompact covering
properties, uniform structures and generalization, dense, uniformspace, complete,
frame, frames
Quaestiones Mathematicae 24(2) 2001, 157-163
every uniformly continuous function on a dense subspace of a uniform space
into a complete uniform space has a unique continuous extension. It is shown
that a uniform frame homomorphism h: L → M from
a complete uniform frame L into a uniform frame M (not necessarily
complete) onto which a dense
surjection ρ: N → M maps has a unique
uniform extension g : L→ N such that ρ ο g = h. We introduce spread uniformities
and uniform spreads into the theory of uniform frames, and, then, show that
the extension so constructed is a uniform spread.
Mathematics Subject Classification (1991): 54D15, 54D20, 54E15
Keywords: uniform frame, uniform local connectedness, spread uniformity,
uniform spread, spread omomorphism, higher separation axioms, noncompact covering
properties, uniform structures and generalization, dense, uniformspace, complete,
frame, frames
Quaestiones Mathematicae 24(2) 2001, 157-163
