By Kotik K Lee
This publication is predicated on a graduate path for scientists and engineers.
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Extra resources for Lectures on Dynamical Systems, Structural Stability, and their Applications
Example text
Then the closure of A, A, is defined to be the set in X consisting of all the points of A along with all the limit points of A. Examples: (1) X= R in the usual topology, A= (xia < x < b}, then A= (xia S x S b}. (2) X= R" and A is the set of points at distance less than r from some fixed point p, then A is the set of points whose distance from p are less than or equal to r. A set A in a topological space X with the property that A = A is called a closed set of X. 7 A set A in a topological space X is closed iff the complement of A in X is an open set.
Then if X is arcwise connected, so is Y. 37 To give the proof is as easy as giving an example, so we opt for the proof. Let p and q be points in Y. Since f is onto, there are points p 1 and q• in X such that f(p')= p and f(q 1 )= q. Since X is arcwise connected, there is a path g (a continuous map) in X joining p 1 and q•. Then the composite map f·g is a continuous map of I into Y such that (f·g) (0)= p and (f·g)(1)= q. ] That is, f·g is a path in Y joining p and q. Since p and q are arbitrary points of Y, the theorem is proved.
The union of all the Sn for n = 1,2, ... is, at most, a denumerable set of points, and sothere is a real number x 0 not belonging to any of the sn. Let V be the (£,x)neighborhood of f 0 for some £ < 1. By the choice of x 0 , f;(X 0 )= 1 for all i, and so none of the members of the sequence f 1 , f 2 , ••• belongs to V, it follows that f 0 cannot be the limit of this sequence. It might be added that the topological space constructed 32 in the above example appears quite naturally and is used frequently in analysis.