Tensor Analysis on Manifolds

ing the description in terms of neighborhoods is not a great chore. If f : X -. Y we say that f is continuous at x e X if for every neighborhood V (<--> a neighborhood) of y = fx there is a neighborhood U (4--). S neighborhood) of x such that U c f V (or f U c V). The following theorem shows that our definition of a continuous function is a correct abstraction of the usual one. Proposition 0.2.6.2. A function f : X Y is continuous iff f is continuous at every x e X. Problem 0.2.6.1. Show that all functions f : X-* X are continuous in the discrete topology and that the only continuous functions in the concrete topology are the constant functions. The notion of a limit can also be abstracted. We define that lima-.xo fx = y if for every neighborhood V of y there is a neighborhood U of xo such that (U {xo}) f-1 V. It follows, as usual, that f is continuous at xo if (a) limx.,xofx = y and (b) fxo = y. A homeomorphism f : X-* Y is a 1-1 onto function such that f and f-1: Y--> X are both continuous. If f : X---> Y is 1-1 but not onto, then f is said to be a homeomorphism into if f and f -1: (range f) -* X are both continuous, where range f is given the relative topology from Y. A homeomorphism f is also called a topological equivalence because f I Tx and f -1 J,, are then 1-1 onto; that is, they give a 1-1 correspondence between the topologies Tx and T, of X and Y. A property of a topological space is said to be a topological property if every homeomorphic space has the property. A topological invariant is a rule which associates to topological spaces an object which is a topological property of the space. The object usually consists of a number or some algebraic system. Problem 0.2.6.2. tan: (7r/2, 7r/2) R is a homeomorphism, where tan = sin/cos. 0.2.7. Connectedness A topological space X is connected if the only subsets of X which are both open and closed arc 0 and X. Another formulation of the same concept, in terms of its negation, is Proposition 0.2.7.1. A topological space X is not connected iff there are nonempty open sets G, H such that G n H = o, G u H = X. A subset A of X is connected if A with the relative topology is connected. The following is not hard to prove. 14 SET THEORY AND TOPOLOGY (Ch.0 Proposition 0.2.7.2. (Chaining Theorem). If {A. I a e J} is a family of connected subsets of X and (l AQ a 0, then U Aa is connected. ad ad) A harder theorem is the following. Proposition 0.2.7.3. If A is connected and A C B, B c A -, then B is connected. In particular, A is connected. The situation for real numbers is particularly simple. An interval (general sense) is a subset of R of one of the forms (a, b)={xIa<x<b}, (a,b]={xIa<x<_b), [a, b) _ {x l a 5 x < b}, [a,b]={xa<_x5b}, where we allow a = -oo, b = oo at open ends, with obvious meanings. The connected sets in R are precisely these intervals. In particular, R itself is connected. Problem 0.2.7.1. Connectedness is a topological property; that is, the image of a connected set under a homeomorphism is connected. Proposition 0.2.7.4. If f : X -+ Y is continuous, and A c X is connected, then fA is connected. In particular, if Y = R, then fA is an interval. (This is a generalization of the intermediate-value theorem for continuous functions of a real variable defined on an interval.) In particular, if f : [a, b] Y is continuous, the range of f is connected. Such an f is called a continuous curve in Y from ya = fa to y, = fb. A topological space Y is arcu'ise connected if for every y,, y, E Y there is a continuous curve from y, to y2. It follows from Propositions 0.2.7.2 and 0.2.7.4 that an arcwise connected space is connected. The connected component of X containing x is the union of all the connected subsets of X which contain x. By Proposition 0.2.7.2 we see that the component containing a is itself connected and that the components containing two different points are either identical or do not meet Thus X is split up into a disjoint union of connected sets, the components of X, each of which is maximal-connected, that is, is not contained in a larger connected set. It follows from Proposition 0.2.7.3 that the components of X are closed. The number of components is a topological invariant. If we substitute "arcwise connected" for "connected" above, we arrive at the notion of the are components of a topological space. The subdivision into arc components is genera;ly finer than the subdivision into components, and §0.2.8] Compactness IS the are components are not necessarily closed. Both these facts are illustrated by the space A in the following example, since A is connected but has two arc components, only one of which is closed. Example. The subset of R2, A= {(x, sin]/x) 1 0< x <_ 1}is connected but not arcwise connected. That it is connected is easy by Proposition 0.2.7.3, since it is the closure of an arcwise connected set B = {(x, sinl/x) 10 < x < 1}. However, the points on the boundary 8B = {(0,y) -1 < y < 1) cannot be joined to those in B by a continuous curve in A. For the open subsets of R" the notions of connectedness and arcwise connectedness coincide. Indeed, in a connected open set of Rn any two points can bejoined by a polygonal continuous curve, that is, a continuous curve for which the range consists of a finite number of straight-line segments. Problem 0.2.7.2. (a) Show that if A is an open set in Rn and a e A, then the set of points in A which can be joined to a by a polygonal continuous curve is an open subset of A. (b) Prove that A is polygonally connected if A is connected.