By Solomon Lefschetz

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B; 1 with 0 < b < 1, The entire interval Œ0; 1. The relative topology is part of the language in this subject, and occurs throughout the rest of the book. We have seen several examples of the relative topology, defined in context, in earlier parts of the book. 4 First Countable, Second Countable, and So Forth There are many different ways to measure the “size” of a set or space. One of these is based on Georg Cantor’s (1845–1918) ideas of cardinality. ✐ ✐ ✐ ✐ ✐ ✐ “topguide” — 2010/12/8 — 17:36 — page 52 — #64 ✐ ✐ 52 2.

It is a remarkable characterization of the circle. 2: First let us show that K has no cut points. p; U0 ; V0 / is a cutting. Then, since U0 [ fpg and V0 [ fpg are both continua, each contains at least some noncut points. Say that y is a noncut point of U0 [ fpg and z is a noncut point of V0 [ fpg. 2 their union K n fy; zg is therefore connected. This contradicts our hypothesis. So K has no cut points. Now, following the statement of the theorem, let p and q be arbitrary distinct points of K. Then K n fp; qg D U [ V , where U and V are nonempty, disjoint, open subsets of K.

We say that a collection W of neighborhoods of x is a neighborhood base (or neighborhood basis) at x if every neighborhood of x contains an element of W. Clearly “neighborhood base” is a local version of the idea of topology. 8. X; U/ be a topological space. We say that X is locally compact if each point of X has a neighborhood base consisting of sets whose closures are compact (such sets are often called precompact). 9. Let X be the real numbers with the usual topology. Let x 2 X. x "; x C "/ for " > 0 form a neighborhood base for the point x, and each of these sets has compact closure.