*Abstract:*

An injective space is a topological space with a strong
extension property for continuous maps with values on it. A certain
filter space construction embeds every *T*_{0} topological space into an injective space. The
construction gives rise to a monad. We show that the monad is of
the Kock-Zöberlein type and apply this to obtain a simple
proof of the fact that the algebras are the continuous lattices
(Alan Day, 1975). In previous work we established an injectivity
theorem for monads of this type, which characterizes the injective
objects over a certain class of embeddings as the algebras. For the
filter monad, the class turns out to consist precisely of the
subspace embeddings. We thus obtain as a corollary that the
injective spaces over subspace embeddings are the continuous
lattices endowed with the Scott topology (Dana Scott, 1972).
Similar results are obtained for continuous Scott domains, which
are characterized as the injective spaces over *dense*
subspace embeddings.

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