Corollary 1. For a stratifiable space following conditions are equivalent:
1) is separable;
2) is weakly separable;
3) is finally compact;
4)has the Souslin property;
5) has countable network;
6) has countable -network.
[1]. Engelking R., General topology, Moscow: Mir, 1986, 752 p.
[2]. Beshimov R.B., On some properties weakly separable spaces, Uzb. Math. Jour. -1994,
№ 1, -С. 7-11.
[3]. Ceder J., Some generalizations of metric spaces, Pacific Journal of Mathematics, 1961, № 1, V 11, -P. 105-126.
[4]. Arhangel’skii A.V., Mappings and spaces, Usp. Mat. Nauk.- 1978, № 4, V 21, C. 133-184.
[5]. Beshimov R.B. Some cardinal invariants and covariant functors in the category of topological spaces and their continuous mappings: Dis. Doct. Phys.-mat. Nauk, Tashkent, 2007.
[6]. Stoltenberg R.A., A note on stratifiable spaces, 2006, p. 294-297.
[7]. Fedorchuk V.V., Filippov V.V., General topology, Basic constructions, Moscow: Fizmatlit, 2006, 332 p.
[8] A.V. Arhangel’skii, Functional tightness, Q-spaces, and τ-embeddings, Comment. Math. Univ. Carol. 24 (1) (1983) 105–119.
Definition 2[8]. Let be an infinite cardinal, and topological spaces. A function is said to be -continuous if for every subspace of such that , the restriction is continuous.
A function is said to be strictly -continuous if for every subspace of X such that , the restriction of to coincides with the restriction to of some continuous function .
The next two statements are immediate from the definition.
Proposition 2.If and are -continuous mappings, then the composition is -continuous. Proposition 3.If is a -continuous mapping, and is a subspace of , then the restriction is -continuous.
