First-countable, Sequential, and Frechet Spaces

Bartlomiej Skorulski

University of Bialystok

Summary.

This article contains a definition of three classes of topological spaces: first-countable, Frechet, and sequential. Next there are some facts about them, that every first-countable space is Frechet and every Frechet space is sequential. Next section contains a formalized construction of topological space which is Frechet but not first-countable. This article is based on [10, pp. 73-81].

MML Identifier: FRECHET

Contents (PDF format)

1. Preliminaries
2. First-countable, Sequential, and {F}rechet Spaces
3. Counterexample of {F}rechet but Not First-countable Space
4. Auxiliary Theorems

Bibliography

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Journal of Formalized Mathematics, 1, 1989.

[2] Grzegorz Bancerek. The ordinal numbers. Journal of Formalized Mathematics, 1, 1989.

[3] Grzegorz Bancerek. Countable sets and Hessenberg's theorem. Journal of Formalized Mathematics, 2, 1990.

[4] Leszek Borys. Paracompact and metrizable spaces. Journal of Formalized Mathematics, 3, 1991.

[5] Czeslaw Bylinski. Functions and their basic properties. Journal of Formalized Mathematics, 1, 1989.

[6] Czeslaw Bylinski. Functions from a set to a set. 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. Families of subsets, subspaces and mappings in topological spaces. Journal of Formalized Mathematics, 1, 1989.

[9] Agata Darmochwal and Yatsuka Nakamura. Metric spaces as topological spaces --- fundamental concepts. Journal of Formalized Mathematics, 3, 1991.

[10] Ryszard Engelking. \em General Topology, volume 60 of \em Monografie Matematyczne. PWN - Polish Scientific Publishers, Warsaw, 1977.

[11] Krzysztof Hryniewiecki. Basic properties of real numbers. Journal of Formalized Mathematics, 1, 1989.

[12] Stanislawa Kanas, Adam Lecko, and Mariusz Startek. Metric spaces. Journal of Formalized Mathematics, 2, 1990.

[13] Beata Padlewska. Families of sets. Journal of Formalized Mathematics, 1, 1989.

[14] Beata Padlewska and Agata Darmochwal. Topological spaces and continuous functions. Journal of Formalized Mathematics, 1, 1989.

[15] Jan Popiolek. Real normed space. Journal of Formalized Mathematics, 2, 1990.

[16] Konrad Raczkowski and Pawel Sadowski. Topological properties of subsets in real numbers. Journal of Formalized Mathematics, 2, 1990.

[17] Andrzej Trybulec. Binary operations applied to functions. Journal of Formalized Mathematics, 1, 1989.

[18] Andrzej Trybulec. Tarski Grothendieck set theory. Journal of Formalized Mathematics, Axiomatics, 1989.

[19] Andrzej Trybulec. Baire spaces, Sober spaces. Journal of Formalized Mathematics, 9, 1997.

[20] Andrzej Trybulec. Subsets of real numbers. Journal of Formalized Mathematics, Addenda, 2003.

[21] Zinaida Trybulec. Properties of subsets. Journal of Formalized Mathematics, 1, 1989.

[22] Edmund Woronowicz. Relations and their basic properties. Journal of Formalized Mathematics, 1, 1989.