General Topology and its Relations to Modern Analysis and by Z. Frolík, M. Katetov, V. Pták

By Z. Frolík, M. Katetov, V. Pták

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Extra resources for General Topology and its Relations to Modern Analysis and Algebra. Proceedings of the Second Prague Topological Symposium, 1966

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2) with hyperplanes of deficiency i orthogonal to the /-dimensional hyperplane spanned by the first i coordinate axes. Let / f be the "natural" inductively defined 1 — 1 function of Si onto /? x I°2 x ... x J?. In a sense, fi + 1 "refines" / f . Then / is the "limit" of t h e / , . Step 3. The embedding / of l2 into s has the (non-obvious) property that s \ fQi) is t n e countable union of compact sets. Hence Corollary 1 is applicable. Thus we have l2 ~ ~l2 ~ / ( / 2 ) ~ 5. 36 R. D. A N D E R S O N The proof of the following theorem uses a classical result of Sierpinski [12] concerning G/s together with some elementary geometry of rc-cells and repeated use of the procedures of Lemmas 1 and 2.

Nöbeling: Grundlagen der analytischen Topologie. Springer, Berlin 1954. 45 PARACOMPACT SUBSETS C. E. AULL Blacksburg In this paper, we distinguish 3 types of paracompact subsets and 2 types of countably paracompact subsets. Definition 1. A subset M of a topological space (X, 3Γ) is ^-paracompact (a-paracompact) if every open cover by members of βΓ has an open locally finite (σ-locally finite) refinement by members of ^ \ Definition 2. A subset M of a topological space (X, ΖΓ) is ot-countably para­ compact if every countable open cover by members of 2Γ has an open locally finite refinement by members of ^".

X -> y is a projective cover in K ¿md g : Z -> Y any essential projection then there exists an essential projection h : X -> Z swc/i that f = h o g. Moreover, if g is also a projective cover then h is a homeomorphism. The last corollary establishes, in particular, the essential uniqueness of projective covers. Also, it states that projective covers, though defined in terms of minimality, are in fact least elements (with respect to a suitably given quasi-order). In a category K which satisfies I — V, the question of the existence of projective objects, and, specifically, of projective covers is settled as follows: Proposition 2.

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