By Solomon Lefschetz

Because the ebook of Lefschetz's Topology (Amer. Math. Soc. Colloquium guides, vol. 12, 1930; observed under as (L)) 3 significant advances have stimulated algebraic topology: the improvement of an summary complicated self sufficient of the geometric simplex, the Pontrjagin duality theorem for abelian topological teams, and the tactic of Cech for treating the homology thought of topological areas by way of platforms of "nerves" each one of that is an summary advanced. the result of (L), very materially extra 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 by way of those new recommendations. A excessive measure of generality is postulated from the outset.

The summary viewpoint with its concomitant formalism allows succinct, distinct presentation of definitions and proofs. Examples are sparingly given, usually of an easy type, which, as they don't partake of the scope of the corresponding textual content, will be intelligible to an hassle-free scholar. yet this is often essentially a ebook for the mature reader, during which he can locate the theorems of algebraic topology welded right into a logically coherent complete

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F is Riemann integrable in R∞ with respect to product measure i∈N μi if and only if for arbitrary positive ε there is a Riemann partition τ ∗ of R∞ such that Sτ ∗ − sτ ∗ < ε. 1 can be obtained by the standard scheme. 1 Define u((xk )k∈N ) = sin(x1−1 ) for (xk )k∈N ∈ (0, 1)∞ . Then u is bounded (by 1) and continuous on (0, 1)∞ , but it is neither uniformly continuous nor continuously extendable to [0, 1]∞ . 1 the following lemma is of some interest. 1 Let f be any bounded and uniformly continuous function on (0, 1)∞ .

Xm ). 11) where 1 is the Lebesgue measure in (0, 1) and ξk ((ωi )i∈N ) = ωk for each k ∈ N and (ωi )i∈N ∈ [0, 1]∞ . Let S be a set of all uniformly distributed sequences on (0, 1). 4 we know that P(S) = 1. The latter relation means that P{ω : (ξk (ω))k∈N is uniformly distributed on the interval (0, 1)} = 1. 12) We put Yn (ω) = (∪nj=1 {ξ j (ω)})m × (ξ1 (ω), ξ1 (ω), . 13) for each n ∈ N . 5 implies that [0,1]m f (x1 , . . , xm )d x1 . . ,n}n lim 45 f (ξi1 (ω), ξi2 (ω), . . , ξin (ω), ξ1 (ω), ξ1 (ω), .

Then f is Riemann integrable on k∈N [ak , bk ] if and only if f is λ almost continuous on k∈N [ak , bk ]. Proof (Necessity) Let f be a Riemann integrable function on k∈N [ak , bk ] ∈ R. 1) k∈I1 where I1 = {k : 1 ≤ k ≤ n & Uk contains at least one inner point p belonging to the set E μ }, where Eμ = x : x ∈ [ak , bk ] & ω( f, x) ≥ μ k∈N and ω( f, x) = lim δ→0 sup x ,x " ∈V (x,δ)∩ k∈N [ak ,bk ] f (x ) − f (x " ) . Here, for x ∈ R∞ and δ > 0, V (x, δ) is denoted by [ak , bk ] &ρ(x, y) ≤ δ . V (x, δ) = y : y ∈ k∈N Because, for k ∈ I1 , p is an inner point of the Uk , there exists V ( p, δ(k, p)) such that V ( p, δ(k, p)) ⊆ Uk .