A Course in Point Set Topology (Undergraduate Texts in by John B. Conway

By John B. Conway

This textbook in element set topology is geared toward an upper-undergraduate viewers. Its light speed should be worthwhile to scholars who're nonetheless studying to write down proofs. necessities contain calculus and at the least one semester of research, the place the coed has been safely uncovered to the information of easy set concept equivalent to subsets, unions, intersections, and capabilities, in addition to convergence and different topological notions within the genuine line. Appendices are integrated to bridge the space among this new fabric and fabric present in an research path. Metric areas are one of many extra wide-spread topological areas utilized in different parts and are hence brought within the first bankruptcy and emphasised through the textual content. This additionally conforms to the procedure of the booklet to begin with the actual and paintings towards the extra normal. bankruptcy 2 defines and develops summary topological areas, with metric areas because the resource of concept, and with a spotlight on Hausdorff areas. the ultimate bankruptcy concentrates on non-stop real-valued services, culminating in a improvement of paracompact areas.

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Additional info for A Course in Point Set Topology (Undergraduate Texts in Mathematics)

Example text

Let x E X and y E X* - X. Suppose that y is defined by a weak star-filter 9 in X which converges to no point in X. If x E U E an,then U 4 9, that is, y 4 U*. Hence y 4 St(x, 4%';). Therefore n{St(x, a:) I a E R} c X, and n{St(x, 43:) I a E R} = a E n} = ci,{x>. n{st(x, an)[ On the other hand, the relation y 4 St(x, 99:) implies that x 4 St( y, 43:). Therefore we have n{St( y , )I %: a E Q} c X * - X. If z E X* - X and z # y, then the weak star-filter Y which defines z is different from 9. ): This proves the lemma.

This shows that y E CI(X* - H). Since H i s open in X*, we have y E X* - Hfor all H E &',which, however, contradicts the assumption that M is an open cover of X*. Finally, (c) is a direct consequence of (b). 0 In concluding this section, we shall show that Shanin's compactification, which is a generalization of the Wallman compactification, is obtained as the completion of a certain generalized uniform space. Let X be a weakly regular space and Y a base for the open sets of X satisfying conditions below: (i) X E Y, (ii) if G, H E Y, then G n H E 9, (iii) if x E G for G E Y, then there exist Gi E Y, i = I , .

Theorem. Any normal cover Q of a space X admits a normal sequence {Q,,} such that Ql < Q and either Card Q, < KOfor each n E N or Card Qn = Card Q for each n E N according as Card Q < KOor KO < Card Q. Proof. 5 and its remark we can inductively construct a sequence {W,,}of open covers such that W: < W n pwhere l , Wo= 42, and either Card Wn< KOfor each n E N or Card W,, = Card Q for each n E N according as Card 42 < KOor KO < Card 9. Let Qn = W2,,for each n E N. Then {Q,,} is the desired normal sequence since we have = W2fn+I) < 0 (Ktn+I)IA < %:+I < Kn = Qn (n > 1).

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