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By Lynn Arthur Steen, J. Arthur Seebach Jr. (auth.)

The inventive strategy of arithmetic, either traditionally and separately, can be defined as a counterpoint among theorems and examples. Al­ even though it'd be dangerous to say that the production of important examples is simpler than the advance of thought, we now have dis­ lined that concentrating on examples is a very expeditious technique of related to undergraduate arithmetic scholars in genuine examine. not just are examples extra concrete than theorems-and therefore extra accessible-but they reduce throughout person theories and make it either acceptable and neces­ sary for the coed to discover the whole literature in journals in addition to texts. certainly, a lot of the content material of this publication was once first defined by way of lower than­ graduate examine groups operating with the authors at Saint Olaf university in the course of the summers of 1967 and 1968. In compiling and enhancing fabric for this e-book, either the authors and their undergraduate assistants discovered a considerable increment in topologi­ cal perception as an immediate results of chasing via info of every instance. we are hoping our readers may have an analogous event. all of the 143 examples during this publication presents innumerable concrete illustrations of definitions, theo­ rems, and common tools of facts. there's no larger manner, for example, to benefit what the definition of metacompactness particularly capability than to attempt to turn out that Niemytzki's tangent disc topology isn't really metacompact. the quest for counterexamples is as vigorous and inventive an job as are available in arithmetic research.

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Example text

A sequence {x .. } in a metric space (X,d) is called a Cauchy sequence iff for every E > 0 there exists an integer N such that d(xm,xn) < E wherever m,n > N. 1). So we define a complete metric space as one in which every Cauchy sequence converges to some point in the space, or equivalently Metric Spaces 37 that the intersection of every nested sequence of closed balls with radii tending to zero is nonempty. ) If the radii do not tend to zero, this condition need not be implied by completeness (Example 135).

Equivalently, X is locally connected if the components of open subsets of X are open in X. Local connectedness clearly does not imply connectedness, but neither does connectedness imply local connectedness (Example 116). However, every hyperconnected space is, clearly locally connected, since in such spaces every open set is connected. Figure 8 summarizes the relevant counterexamples. Connected (116) Path connected (121) Arc connected (120) r--- ----I I I I I I (45) ---- IL (46) I Hyperconnected (53) (56) I I - - - - i- (57) (7) (18) --, I I I I I I I I LocaUy _____________ _ _connected _ _ _ JI Figure 8.

X is separable, since {p I is a countable dense subset. But, if X is uncountable, X - {p I is not separable. 7. If X is uncountable, it is first countable, but not second countable, since X - {p I is discrete. 8. If on a given set X, we define Tl to be the collection of all sets containing a point p, and T2 to be the collection of all sets containing q ¢ p, the spaces (X,Tl) and (X,T2) are homeomorphic, but Tl and T2 are not comparable. 9. X is scattered, since every subset not containing p has no limit point, and for a subset which contains p, p itself is not a limit point.

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