By Man Y.

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**Extra resources for Introduction to topology: Theory and applications**

**Example text**

What about the maps A → A ∩ E, A → A − E and A → E − A? 2. Prove that f (A) = {2a : a ∈ A} and g(A) = {a : 2a ∈ A} are continuous. 3. The union, intersection and difference can be considered as maps from P(N)2 to P(N). Are these maps continuous? 4. Prove that 8 <0, if A = ∅ : P(N) → Qusual 1 , if A = ∅ min A is continuous. What if the metric on Q is the p-adic one? 6. Prove that if f, g : X → Rusual are continuous, then f + g and f g are also continuous. 7. Prove that if f : X → Y and g : Z → W are continuous, then the map h(x, z) = (f (x), g(z)) : X × Z → Y × W is continuous.

Is the map still continuous if the metric dX on X is changed to another metric dX satisfying dX (x1 , x2 ) ≤ c dX (x1 , x2 ) for a constant c > 0? What if dX satisfies dX (x1 , x2 ) ≤ c dX (x1 , x2 ) instead? What if the metric on Y is also similarly modified? 10. Fix a point a in a metric space X. Prove that f (x) = d(x, a) : X → Rusual is continuous. 4. 11. Let X be a metric space with distance d. 13. Prove that d : X × X → Rusual is a continuous map. 12. A sequence {an : n ∈ N} in a metric space X is said to have limit x (or converges to x) and denoted lim an = x, if for any > 0, there is N , such that n > N implies d(an , x) < .

Prove that d -open subsets are d-open. In particular, if two metrics are equivalent: c1 d(x, y) ≤ d (x, y) ≤ c2 d(x, y) for some constants c1 , c2 > 0, then d-open is equivalent to d -open. Two metrics are said to induce the same topology if the openness in one metric is the same as the openness in the other. 10. 3 induce the same topology. 11. Prove that in C[0, 1], L1 -open implies L∞ -open, but the converse is not true. 36 Chapter 2. 12. Suppose d and d are two metrics on X. Prove the following are equivalent.