By Carlos R Borges

The cloth during this publication is geared up in this kind of means that the reader will get to major purposes speedy, and the emphasis is at the geometric figuring out and use of recent thoughts. The topic of the booklet is that topology is admittedly the language of recent arithmetic.

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**Extra resources for Elementary Topology and Applications**

**Example text**

Then, by (v), ,f-' ( Y -U)- c f - ' ( Y - U ) which implies that f - '(Y -U)- = f' (Y - U ) . Therefore, p E [ X - f -I ( Y - U ) ] E z and f ( X - f - ' (Y - U ) ) = U . (ii) implies (vi): Obvious. consists of unions of commutes with unions and (vi) implies (ii): Straightforward,since the topology finite intersections of elements of z and f intersections. IS. Corollary. ~f continuous. x ' :Y -' CT > Z and f and g are continuous, then g o f is Proof. Immediate from Theorem 14 (ii), because 16. Corollary.

6. Definition. Given a space X, finite families { q I i = 1,. . , n } of spaces and functions f i : X + y,, gi : 6 4 Zi , i = 1,. . ) 51 From Old to New Spaces Because of Theorem 3, the following result is immediate. At this stage, the reader may wonder: But it seems that the definition of the product topology could be applied to infinite products verbatim. Can it? Of course, it can! Furthermore, it is easy to see that all results of this section are valid for infinite products. Indeed, with the possible exception of Lemma 5 , the proofs of the other results apply to infinite products verbatim.

Lemmas 2 , 4 and 7 and Theorem 3 remain valid for any product spaces. , n , be a finite family of metric spaces. By analogy with the standard definition of distance i n the Cartesian plane ( i . e . , d ( ( x , , xz), ( y , , y 2 ) ) = J(x, - Y , ) +~ ( x , - y2)2 ), we define a function d : ( 1 7 ~ = , X i ) x ( 1 7 ~ = ,+ X i El ) by This function d will always be referred to as the product-metric. 9. Proposition. The product-metric d is actually a metric on I-Iy=, X i . Proof. Certainly, we only need to verify the triangle inequality.