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By Sylvester J.J.

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Now define an e-neighborhood U of x E X as U =(y e xl d(x,y) < e}. Then define a neighborhood of x eX to be any set in X containing an e-neighborhood of x. Prove that with these definitions, X becomes a topological space. Remark: A topological space defined in such a way is called a metric space. Also, prove that Euclidean space is a metric space. 2 27 If A and B are any sets, then the set of all pairs (a,b) with a e A and b e B is denoted by A x B, and is called the product set of A and B. Let X and Y be two topological spaces.

Up to now, we have been considering the topological space, its subsets and their properties. In the following, we shall briefly discuss the relationships between spaces and the topological properties of spaces. Let X and Y be two topological spaces. A mapping f of X into Y is a rule which assigns to each point p E X a well defined point f(p) in Y. The mappping f is continuous at p if, for each neighborhood U of f(p) in Y, there is a neighborhood V of p in X such that f(V) c u. The mapping f of X into Y is continuous if it is continuous at all points of X.

A topological space X is a Hausdorff space, sometimes denoted by T2 space, if for every pair p, q e X with p q, there is a neighborhood U of p and a neighborhood v of q such that U n V = o. 5 Let X be a Hausdorff space, and suppose that a sequence (Pn} has a limit p. Then this limit is unique. Example: R1 is T2• In general, R" in its usual topology is a T2 space. Let A be a set of points in a topological space X. Then a point p E X is a limit point (or accumulation point) of A if every neighborhood of p contains a point of A different from p.

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