By Waclaw Sierpiński

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A subset A of a space E is said to be connected if the subspace A of E is connected. EXAMPLE 1. We shall prove later, in the study of metric spaces, that R (as well as every interval of R) is connected; we shall for the moment assume this. EXAMPLE 2. On the other hand, the set Q of rationals is not connected; more generally, we shall show that if a subset A of R is not an interval, it is not connected. I n fact, there then exist two distinct points x, y E A such that [x, y ] $ A; therefore there exists a point a E [x, y ] such that a $ A.

WE THEN WRITE limaf = b. @ CONVERGES T O b. 6. Definition. LETf SPACE BE A FILTER BASE ON Y; LET BE A MAPPING OF A SET TOPOLOGICAL SPACES AND METRIC SPACES 26 [Ch. I EXAMPLE 1. Let (a,) be a sequence of points of Y; we denote by f the mapping n -+ a, of N into Y, and by A? the collection of subsets of N whose complements are finite. 1. EXAMPLE 2. If X is a topological space and 9’ denotes the collection of neighborhoods of a point a of X, to say that f(a) is the limit off along 9’ is equivalent to saying that f is continuous at the point a.

Product of subspoces If A, denotes a subspace of Ei , one can verify that the product topology on A = IIAi is identical with the topology induced by that of n E i on its subset A. , a,). Associativity of the topological product. If A, B, and C are topological spaces, the one-to-one canonical correspondence between the spaces (A x B) x C (respectively, A x (B x C)) and A x B x C is a homeornorplzisrn. It suffices (see Section 5 ) to show that this correspondence preserves neighborhoods. But every point ( a , b, c) of (A x B) x C has a neighborhood base consisting of the sets (mu x Wb) x wc ; and every point ( a , b, c) of A x B x C has a neighborhood base consisting of the sets wcLx wb x w, (where w , , w b , w , denote open neighborhoods of a, b, c in A, B, C).