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Example text

We set −ri . This generalized sum fulfills the usual rules for addition and gives a finite sum the same value as the ordinary sum. Let A be a set. We define ℓ2A = (xα )α∈A ∈ RA : α∈A x2α < ∞ . By the Schwarz Inequality, we have, for all (xα ), (yα ) ∈ ℓ2A , that 2 α∈E |xα yα | ≤ for every finite E ⊂ A. Hence we have that α∈E α∈A x2α · yα2 α∈E |xα yα | < ∞. As a consequence, we can define a function (, . , ) : ℓ2A × ℓ2A → R by the formula (xα ), (yα ) = xα y α . α∈A 48 By the foregoing, we have for all (xα ), (yα ) ∈ ℓ2A that (xα + yα )2 = α∈A (x2α + 2xα yα + yα2 ) = α∈A x2α + 2 (xα ), (yα ) + α∈A α∈A yα2 < ∞ .

Let d be a pseudometric on X. For all x ∈ X and r ≥ 0, we set Bd (x, r) = {y ∈ X : ¯d (x, r) = {y ∈ X : d(y, x) ≤ r}; These sets are called the open ball of d(y, x) < r} and B radius r around x and the closed ball of radius r around x, respectively. For every pseudometric d on X, the family {Bd (x, r) : x ∈ X and r > 0} is a base for a topology of X; we denote this topology by τd and we call it the topology induced by d. Let X be a topological space and d pseudometric on X. We say that d is a continuous pseudometric of X if d is continuous as a mapping from the product space X 2 to R.

The family C = {gi−1 (O) : i ∈ I ja O ⊂◦ R} is a subbase of the topology τ . To prove the inclusion τ ⊂ τd , it suffices to show that C ⊂ τd . Let i ∈ I, O ⊂◦ R and x ∈ gi−1 (O). Since gi (x) ∈ O ⊂◦ R, there exists r > 0 such that (g(x) − r, g(x) + r) ⊂ O. Now we have that Bd (x, r) ⊂ gi−1 (O), because if d(y, x) < r, then |gi (y) − gi (x)| < r and hence gi (y) ∈ (g(x) − r, g(x) + r) ⊂ O, which implies that y ∈ gi−1 (O). By the foregoing, we have that gi−1 (O) ∈ τd . We know (from a first course) that a metric space (X, d) can be isometrically embedded in the Banach space ℓ∞ X (this is the linear space of all bounded functions X → R, equipped with the supremum-norm).

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A second course in general topology by Heikki Junnila


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