By Iain T. Adamson

This paintings goals to give easy topology in an unconventional means. It offers a evaluation of the fundamental definitions including workouts with out recommendations or proofs of the theorems partially 1, after which offers the suggestions partly 2, permitting the coed to check solutions with their very own.

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Since Fn is the smallest sequentially closed subset containing An , we obtain Fn ⊆ F. But x ∈ n∈N Fn , so x ∈ F, d . 12). 8 (Tychonoff, metric case, finite products) Let X i , di be finitely many metric spaces, 1 i m. Equip the product space X = X 1 × · · · × X m with the metric d defined by d((x1 , . . , xm ), (y1 , . . , ym )) = m maxi=1 di (xi , yi ). Every product K 1 × · · · × K m of sequentially compact subsets K i of X i , 1 i m, is sequentially compact in X . Proof Write m-tuples x as (x1 , .

5 Open balls in R2 . ball with center x and radius d(x, y) < . 2 In R with distance |x − y|, the open ball of center x and radius r is the open interval (x − r, x + r ). 5, for various metrics (the gray areas, without the boundaries). 3 (Sequentially closed) Let X, d be a metric space. For every subset F of X , the following are equivalent: 1. F is sequentially closed; d centered 2. for every point x ∈ X that is not in F, there is an open ball Bx,< at x that does not meet F; 3. the complement of F is a union of open balls.

For every open subset U of X , U meets cl(A) if and only if U meets A.