By Solomon Lefschetz

Because the booklet of Lefschetz's Topology (Amer. Math. Soc. Colloquium guides, vol. 12, 1930; observed under as (L)) 3 significant advances have inspired algebraic topology: the advance of an summary advanced autonomous of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the strategy of Cech for treating the homology conception of topological areas via structures of "nerves" each one of that's an summary complicated. the result of (L), very materially additional to either by means of incorporation of next released paintings and through new theorems of the author's, are right here thoroughly recast and unified when it comes to those new ideas. A excessive measure of generality is postulated from the outset.

The summary perspective with its concomitant formalism allows succinct, specific presentation of definitions and proofs. Examples are sparingly given, more often than not of an easy style, which, as they don't partake of the scope of the corresponding textual content, may be intelligible to an undemanding scholar. yet this can be basically a publication for the mature reader, during which he can locate the theorems of algebraic topology welded right into a logically coherent entire

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Example text

A subset F of X is closed if and only if given any point x in X — F, D(x, F) ^ 0. Proof. If F is a closed subset of X, then X — F is open. Therefore, given any x E l — F, there is a positive number p such that N(x, p) C X — F. But then D(x, F) > p; hence D(x, F) 5^ 0. Suppose, on the other hand, that given any point x in X — F r D(x, F) 0. Then setting p = D(x, F), we have N(z, p) C X — F. That is, for each x e X — F, we have a positive number p such that N(:r, p) C X — Fj which is to say that X — F is open.

Let D i and D 2 b e p ossib le m etrics for a set X. D\ an d D 2 are said t o be equivalent if e v e ry D i- o p e n set is D 2-open and ev e ry D 2-open set is D i-open . P rov e th at D 1 an d D 2 are eq u iv a len t if and on ly if, giv en an y x G X and an y p > 0, there are p o sitiv e n u m bers p i an d P 2 su ch that the D i- p ^ n e ig h ­ b o rh o o d o f x is a su b se t o f the D 2-p-n eigh borh ood o f x, and the D 2-P2n e igh b o rh ood o f x is a su b se t of th e D i-p -n eigh borh ood of x.

By Definition 5(iii), there is N3 E 9lx such that x e N 3c N 1n N 2c U n V ; hence U n V is also an open set. open sets is again an open set. The intersection of any two iii) Suppose {Ui}, i E /, is a family of open sets, and x E U j Ui. Then x E Ui for some i; hence there is N E 9lx such that x G N C [/f c U / The union of any family of open sets is thus again an open set. Therefore r is a topology on X. The reader should compare the proof of Proposition 7 with the proof of Proposition 2, Chapter 2.

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