Journal of Formalized Mathematics
Volume 8, 1996
University of Bialystok
Copyright (c) 1996 Association of Mizar Users

## The Way-Below'' Relation

Grzegorz Bancerek
Warsaw University, Bialystok

### Summary.

In the paper the way-below" relation, in symbols $x \ll y$, is introduced. Some authors prefer the term relatively compact" or way inside", since in the poset of open sets of a topology it is natural to read $U \ll V$ as $U$ is relatively compact in $V$". A compact element of a poset (or an element isolated from below) is defined to be way below itself. So, the compactness in the poset of open sets of a topology is equivalent to the compactness in that topology.\par The article includes definitions, facts and examples 1.1-1.8 presented in [11, pp. 38-42].

This work has been partially supported by Office of Naval Research Grant N00014-95-1-1336.

#### MML Identifier: WAYBEL_3

The terminology and notation used in this paper have been introduced in the following articles [15] [19] [20] [10] [6] [7] [16] [1] [18] [17] [14] [21] [9] [8] [5] [2] [3] [12] [4] [13]

#### Contents (PDF format)

1. The Way-Below'' Relation
2. The Way-Below Relation in Other Terms
3. Continuous Lattices
4. The Way-Below Relation in Direct Powers
5. The Way-Below Relation in Topological Spaces

#### Bibliography

[1] Grzegorz Bancerek. K\"onig's theorem. Journal of Formalized Mathematics, 2, 1990.
[2] Grzegorz Bancerek. Complete lattices. Journal of Formalized Mathematics, 4, 1992.
[3] Grzegorz Bancerek. Bounds in posets and relational substructures. Journal of Formalized Mathematics, 8, 1996.
[4] Grzegorz Bancerek. Directed sets, nets, ideals, filters, and maps. Journal of Formalized Mathematics, 8, 1996.
[5] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.
[6] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[7] Czeslaw Bylinski. The modification of a function by a function and the iteration of the composition of a function. Journal of Formalized Mathematics, 2, 1990.
[8] Agata Darmochwal. Compact spaces. Journal of Formalized Mathematics, 1, 1989.
[9] Agata Darmochwal. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.
[10] Agata Darmochwal. Finite sets. Journal of Formalized Mathematics, 1, 1989.
[11] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, and D.S. Scott. \em A Compendium of Continuous Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1980.
[12] Adam Grabowski and Robert Milewski. Boolean posets, posets under inclusion and products of relational structures. Journal of Formalized Mathematics, 8, 1996.
[13] Artur Kornilowicz. Meet -- continuous lattices. Journal of Formalized Mathematics, 8, 1996.
[14] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.
[15] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.
[16] Andrzej Trybulec. Many-sorted sets. Journal of Formalized Mathematics, 5, 1993.
[17] Wojciech A. Trybulec. Partially ordered sets. Journal of Formalized Mathematics, 1, 1989.
[18] Wojciech A. Trybulec. Groups. Journal of Formalized Mathematics, 2, 1990.
[19] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.
[20] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.
[21] Miroslaw Wysocki and Agata Darmochwal. Subsets of topological spaces. Journal of Formalized Mathematics, 1, 1989.