By Michael Henle

Very good textual content for upper-level undergraduate and graduate scholars exhibits how geometric and algebraic rules met and grew jointly into an immense department of arithmetic. Lucid assurance of vector fields, surfaces, homology of complexes, even more. a few wisdom of differential equations and multivariate calculus required. Many difficulties and routines (some strategies) built-in into the textual content. 1979 variation. Bibliography.

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Extra resources for A Combinatorial Introduction to Topology (Dover Books on Mathematics)

Sample text

We suppose that V is defined not only at P but in some neighborhood of P. If P is an ordinary point, then V(P) is not zero and it is possible to choose the neighborhood so that V is never zero there. If P is a critical point, then it may still be possible to choose a neighborhood so that V vanishes only at P in the neighborhood. Then P is called an isolated critical point. 1 are isolated.

A vector field Fon a subset D of the plane is a function assigning to each point P of D a vector in the plane with its tail at P. Intuitively we can think of V as giving the velocity of some substance that is presently in D but in a state of agitation, like water in a bathtub. Placement of the vector V(P) with its tail at P is mainly for dramatic effect, to aid in visualizing the vector field. 1 A vector field. essential qualities of the vector V(P) are its length and direction. For practical purposes it is usually more convenient to place all vectors with their tails at the origin.

To prove this, let u: D -• R be a topological transformation from a set D with the fixed point property to a set R. Let g be a continuous transformation of R into R. To find a fixed point for g, consider the composition / = v o g o w, where i; is the inverse transformation v:R^> D. This composition is a continuous transformation of D to D. By hypothesis, / has a fixed point P in D, v(g(u{P))) = f(P) — P. This means that g(u(P)) = u(P\ so that u(P) is a fixed point for #. This proves that R has the fixed point property.