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.

#### MML Identifier: WAYBEL11

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]

#### Contents (PDF format)

1. Preliminaries
2. Scott Topology
3. Scott Convergence

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