Scott Topology
Journal of Formalized Mathematics
Volume 9, 1997
University of Bialystok
Copyright (c) 1997
Association of Mizar Users
Scott Topology
-
Andrzej Trybulec
-
Warsaw University, Bialystok
Summary.
-
In the article we continue the formalization in Mizar of [12, 98-105].
We work with structures of the form
$$L = \langle C,\ \leq,\ \tau \rangle,$$
where $C$ is the carrier of the structure,
$\leq$ - an ordering relation on $C$ and
$\tau$ a family of subsets of $C$.
When $\langle C,\ \leq \rangle$ is a complete lattice
we say that $L$ is Scott, if $\tau$ is the Scott topology
of $\langle C,\ \leq \rangle$.
We define the Scott convergence (lim inf convergence).
Following [12] we prove that
in the case of a continuous lattice $\langle C,\ \leq \rangle$
the Scott convergence
is topological, i.e. enjoys the properties: (CONSTANTS), (SUBNETS),
(DIVERGENCE), (ITERATED LIMITS). We formalize the theorem, that if
the Scott convergence has the (ITERATED LIMITS) property,
the $\langle C,\ \leq \rangle$
is continuous.
This work was partially supported by
the Office of Naval Research Grant N00014-95-1-1336.
The terminology and notation used in this paper have been
introduced in the following articles
[23]
[10]
[29]
[31]
[11]
[30]
[7]
[9]
[8]
[2]
[28]
[19]
[21]
[32]
[22]
[20]
[34]
[24]
[1]
[18]
[27]
[3]
[4]
[5]
[13]
[33]
[14]
[15]
[16]
[6]
[25]
[17]
[26]
-
Preliminaries
-
Scott Topology
-
Scott Convergence
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Received January 29, 1997
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