Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Formalized Mathematics

(a computer assisted approach)

Editor-in-Chief: Matuszewski, Roman

4 Issues per year


SCImago Journal Rank (SJR) 2016: 0.207
Source Normalized Impact per Paper (SNIP) 2016: 0.315

Open Access
Online
ISSN
1898-9934
See all formats and pricing
More options …
Volume 17, Issue 3 (Jan 2009)

Issues

Basic Properties of Metrizable Topological Spaces

Karol Pąk
Published Online: 2010-07-08 | DOI: https://doi.org/10.2478/v10037-009-0024-8

Basic Properties of Metrizable Topological Spaces

We continue Mizar formalization of general topology according to the book [11] by Engelking. In the article, we present the final theorem of Section 4.1. Namely, the paper includes the formalization of theorems on the correspondence between the cardinalities of the basis and of some open subcover, and a discreet (closed) subspaces, and the weight of that metrizable topological space. We also define Lindelöf spaces and state the above theorem in this special case. We also introduce the concept of separation among two subsets (see [12]).

  • [1] Grzegorz Bancerek. Cardinal arithmetics. Formalized Mathematics, 1(3):543-547, 1990.Google Scholar

  • [2] Grzegorz Bancerek. Cardinal numbers. Formalized Mathematics, 1(2):377-382, 1990.Google Scholar

  • [3] Grzegorz Bancerek. König's theorem. Formalized Mathematics, 1(3):589-593, 1990.Google Scholar

  • [4] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.Google Scholar

  • [5] Józef Białas and Yatsuka Nakamura. The theorem of Weierstrass. Formalized Mathematics, 5(3):353-359, 1996.Google Scholar

  • [6] Leszek Borys. Paracompact and metrizable spaces. Formalized Mathematics, 2(4):481-485, 1991.Google Scholar

  • [7] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.Google Scholar

  • [8] Agata Darmochwał. Families of subsets, subspaces and mappings in topological spaces. Formalized Mathematics, 1(2):257-261, 1990.Google Scholar

  • [9] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.Google Scholar

  • [10] Agata Darmochwał. The Euclidean space. Formalized Mathematics, 2(4):599-603, 1991.Google Scholar

  • [11] Ryszard Engelking. General Topology, volume 60 of Monografie Matematyczne. PWN-Polish Scientific Publishers, Warsaw, 1977.Google Scholar

  • [12] Ryszard Engelking. Teoria wymiaru. PWN, 1981.Google Scholar

  • [13] Adam Grabowski. Properties of the product of compact topological spaces. Formalized Mathematics, 8(1):55-59, 1999.Google Scholar

  • [14] Adam Grabowski. On the Borel families of subsets of topological spaces. Formalized Mathematics, 13(4):453-461, 2005.Google Scholar

  • [15] Adam Grabowski. On the boundary and derivative of a set. Formalized Mathematics, 13(1):139-146, 2005.Google Scholar

  • [16] Stanisława Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Formalized Mathematics, 1(3):607-610, 1990.Google Scholar

  • [17] Zbigniew Karno. Maximal discrete subspaces of almost discrete topological spaces. Formalized Mathematics, 4(1):125-135, 1993.Google Scholar

  • [18] Robert Milewski. Bases of continuous lattices. Formalized Mathematics, 7(2):285-294, 1998.Google Scholar

  • [19] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.Google Scholar

  • [20] Alexander Yu. Shibakov and Andrzej Trybulec. The Cantor set. Formalized Mathematics, 5(2):233-236, 1996.Google Scholar

  • [21] Andrzej Trybulec. A Borsuk theorem on homotopy types. Formalized Mathematics, 2(4):535-545, 1991.Google Scholar

  • [22] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.Google Scholar

  • [23] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.Google Scholar

About the article


Published Online: 2010-07-08

Published in Print: 2009-01-01


Citation Information: Formalized Mathematics, ISSN (Online) 1898-9934, ISSN (Print) 1426-2630, DOI: https://doi.org/10.2478/v10037-009-0024-8.

Export Citation

This content is open access.

Citing Articles

Here you can find all Crossref-listed publications in which this article is cited. If you would like to receive automatic email messages as soon as this article is cited in other publications, simply activate the “Citation Alert” on the top of this page.

[1]
Marco Riccardi
Formalized Mathematics, 2012, Volume 20, Number 1
[2]
Marco Riccardi
Formalized Mathematics, 2011, Volume 19, Number 1
[3]
Karol Pąk
Formalized Mathematics, 2009, Volume 17, Number 3

Comments (0)

Please log in or register to comment.
Log in