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The 2-sphere is the standard sphere, and the 1-sphere is just a circle. We shall revisit the interesting topological properties of the 3-sphere in subsequent chapters. 7. ) |OJ £ {0,1} foralU}. ,^ = {a e X | cii — Ci for all 0 < i < n}. Define a subset of X to be open if it is a union of cylinders. Intuitively, two sequences are "close" if they agree in the first several entries. This apparently odd space demonstrates the potential for very abstract spaces. However, even though the definition looks abstract, it appears in numerous applications.

B then B is infinite. 8 Prove that the union of an uncountable set with a countable one is uncountable. 9 Prove that the union of countably many countable sets is countable. 10 Prove that the product of countably many countable sets is uncountable, Proposition 5. ] xxxvn Topology and Its Applications by William F. Basener Copyright © 2006 John Wiley & Sons, Inc. 1 CONTINUITY AND OPEN SETS IN RN Continuity is the foundation on which topology is built. We show in this chapter that continuity relies solely on open sets.

Basener Copyright © 2006 John Wiley & Sons, Inc. 1 CONTINUITY AND OPEN SETS IN RN Continuity is the foundation on which topology is built. We show in this chapter that continuity relies solely on open sets. Focus on these most basic notions, continuity and open sets, lends elegance and generality to the theory of topology. Moreover, because topology is built on continuity, applications of topology often depend only on "continuous type of information" and not necessarily "distance-angle-derivative type of information," which makes topology a particularly powerful tool.