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Extra info for An introduction to electrodynamics from the standpoint of the electron theory
E. with all tangent vectors making non-zero angles with each other. In order to compute the index of p with respect to an immersed curve f , let us join p by a (nonclosed) smooth curve α transversal to f to a far away point a and move from a to p along that curve. , so that the tangent vector to f looks to the right of α) and subtract one when we cross it in the negative direction. When we reach the connected component of the complement to the curve containing p, we will obtain a certain integer i(p).
Define F on the interval I as the increasing linear map whose image is the interval [1/3, 2/3]. Consider the longest intervals I1 and I2 to the right and to the left to I. 2/9] and [7/9, 8/9], respectively. 2/9] ∪ [1/3, 2/3] ∪ [7/9, 8/9] and so on by induction. Eventually one obtains a strictly monotone bijective map [m, M ] \ A → [0, 1] \ C which by continuity is extended to the desired homeomorphism. 3. Prove that the product of countably many finite sets with the discrete topology is homeomorphic to the Cantor set.
Since local degree can be calculated at any non-critical value this shows that d(f ) = d(fP L ) = d(Edeg f ) = deg f. 5. , there exists a p ∈ Dn such that f (p) = p. The simplest instance of this theorem (for n = 1) is an immediate corollary of the intermediate value theorem from calculus since a continuous map f of a closed interval [a, b] into itself can be considered as a real-values function such that f (a) ≥ a and f (b) ≥ b. Hence by the intermediate value theorem the function f (x) − x has a zero on [a, b].