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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Received January 29, 1997


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