First, equivalence conditions for connectedness are examined for a finite topological space (originated in [9]). Secondly, definitions of subspace, and components of the subspace of a finite topological space are given. Lastly, concepts of continuous finite sequence and minimum path of finite topological space are proposed.
[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
[2] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
[3] Czesław Byliński. Functions and their basic properties. Formalized Mathematics, 1(1):55-65, 1990.
[4] Czesław Byliński. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
[5] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
[6] Hiroshi Imura and Masayoshi Eguchi. Finite topological spaces. Formalized Mathematics, 3(2):189-193, 1992.
[7] Hiroshi Imura, Masami Tanaka, and Yatsuka Nakamura. Continuous mappings between finite and one-dimensional finite topological spaces. Formalized Mathematics, 12(3):381-384, 2004.
[8] Jarosław Kotowicz. Functions and finite sequences of real numbers. Formalized Mathematics, 3(2):275-278, 1992.
[9] Yatsuka Nakamura. Finite topology concept for discrete spaces. In H. Umegaki, editor, Proceedings of the Eleventh Symposium on Applied Functional Analysis, pages 111-116, Noda-City, Chiba, Japan, 1988. Science University of Tokyo.
[10] Takaya Nishiyama and Yasuho Mizuhara. Binary arithmetics. Formalized Mathematics, 4(1):83-86, 1993.
[11] Beata Padlewska and Agata Darmochwał. Topological spaces and continuous functions. Formalized Mathematics, 1(1):223-230, 1990.
[12] Masami Tanaka and Yatsuka Nakamura. Some set series in finite topological spaces. Fundamental concepts for image processing. Formalized Mathematics, 12(2):125-129, 2004.
[13] Andrzej Trybulec. Subsets of complex numbers. To appear in Formalized Mathematics.
[14] Andrzej Trybulec. Domains and their Cartesian products. Formalized Mathematics, 1(1):115-122, 1990.
[15] Andrzej Trybulec. Enumerated sets. Formalized Mathematics, 1(1):25-34, 1990.
[16] Andrzej Trybulec. Tarski Grothendieck set theory. Formalized Mathematics, 1(1):9-11, 1990.
[17] Wojciech A. Trybulec. Pigeon hole principle. Formalized Mathematics, 1(3):575-579, 1990.
[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1(1):73-83, 1990.