By Alejandro Illanes

ISBN-10: 0824719824

ISBN-13: 9780824719821

Offers hyperspace basics, delivering a easy evaluate and a origin for extra examine. themes comprise the topology for hyperspaces, examples of geometric versions for hyperspaces, 2x and C(X) for Peano continua X, arcs in hyperspaces, the form and contractability of hyperspaces, hyperspaces and the fastened aspect estate, and Whitney maps. The textual content comprises examples and routines all through, and offers proofs for many effects.

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Example text

26 Exercise. , S’ = {(z,y) E R2 : x2 +y2 = 1)). Then F2(S1) is a Moebius band; that is, Fz(S’) is homeomorphic to the quotient space obtained from [0, l] x [0, l] by identifying the point (0,~) with the point (1,l - y) for each 9 E [0, 11. Remark. Some of the results in the exercises above are in [6]. If you read [6], you should be aware of two errors: (c) on p. 3 on p. 162 (cf. 5 of [7, pp. 40-411 and [8 or 91, respectively). You should also be careful to remember the standing assumption on p.

Polon. Sci. Cl. R. 19 (1958), 668. 3. 4. F. , Berlin, 1927. K. Kuratowski, Topology, Vol. I, Acad. , 1966. 5. 6. K. Kuratowski, Topology, Vol. II, Acad. , 1968. Ernest Michael, Topologies on spacesof subsets,Trans. Amer. Math. Sot. 71 (1951), 152-182. REFERENCES 7. 8. 9. 29 Sam B. , Vol. , 1978. E. Smithson, First countable hyperspaces, Proc. Amer. Math. Sot. 56 (1976), 325-328. Daniel E. Wulbert, Subsets of first countable spaces, Proc. Amer. Math. Sot. 19 (1968), 1273-1277. This Page Intentionally Left Blank II.

If Y is an infinite, discrete space, then the Vietoris topology for CL(Y) does not have a countable base. Let ,0 be a base for the Vietoris topology for CL(Y). Note that Proof. each A E CL(Y) is open in Y. A C (A). Thus, it follows easily that (1) A = USA for each A E CL(Y). Now, it follows immediately from (1) that if A, A’ E CL(Y) such that A # A’, then DA # a,&. jA is a one-toone function from CL(Y) into p. Hence, A RESULT ABOUT METRIZABILITY c-4 ICW)l OF CL(X) 13 I IPI- Since Y is a discrete space, CL(Y) = {A c Y : A # 0}; thus, since Y is an infinite set, CL(Y) is uncountable.

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Hyperspaces: Fundamentals and Recent Advances by Alejandro Illanes

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