Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.

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topology question6
Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a function
defined by
dp(x, y) = p-max(mɛNp™ divides (2-3))
Prove that d, is a metric on Z and that dp(x, y) < max{dp(x, 2), dp(z,y)}, for every x, y.
ZE Z
Exercise 2. [Closed in metric space] Let (X, d) be a metric space and F € X be a
finite subset. Prove that F is closed in X.
Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty
subset of X. The distance of a point X from the subset Y is a function X→ [0, +∞[
defined by
d(x, y) = inf{d(x, y); y = Y}.
1. Verify that the distance function is well defined.
2. Prove that Y = {x € X; d(x, y) = 0}.
Exercise 4. [Separable space] Let X be a set of all real sequences (zn)neN converging
to 0. Prove that the function
d: X X X → [0, +∞o[
(In, Yn) → d(In, Yn) = sup |In - yn
NEN
is a metric on X. Show that the metric space (X, d) is separable.
Exercise 5. [Restriction of metric] Let (X,d) be a metric space and U be a proper
open subset of X. Consider a function
du : U x U→ [0, +∞[
1
d(x, X\U
Prove that du is a metric on U and that it is equivalent to the induced metric duxu.
(x,y) → dv(x, y) = d(x, y) +
1
d(y, X\U
Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}.
Prove that is a topology on X.
Transcribed Image Text:Exercise 1. [Metric] Let p be a prime number, and d: ZxZ→ [0, +∞[ be a function defined by dp(x, y) = p-max(mɛNp™ divides (2-3)) Prove that d, is a metric on Z and that dp(x, y) < max{dp(x, 2), dp(z,y)}, for every x, y. ZE Z Exercise 2. [Closed in metric space] Let (X, d) be a metric space and F € X be a finite subset. Prove that F is closed in X. Exercise 3. [Closure in metric space] Let (X, d) be a metric space and Y be a nonempty subset of X. The distance of a point X from the subset Y is a function X→ [0, +∞[ defined by d(x, y) = inf{d(x, y); y = Y}. 1. Verify that the distance function is well defined. 2. Prove that Y = {x € X; d(x, y) = 0}. Exercise 4. [Separable space] Let X be a set of all real sequences (zn)neN converging to 0. Prove that the function d: X X X → [0, +∞o[ (In, Yn) → d(In, Yn) = sup |In - yn NEN is a metric on X. Show that the metric space (X, d) is separable. Exercise 5. [Restriction of metric] Let (X,d) be a metric space and U be a proper open subset of X. Consider a function du : U x U→ [0, +∞[ 1 d(x, X\U Prove that du is a metric on U and that it is equivalent to the induced metric duxu. (x,y) → dv(x, y) = d(x, y) + 1 d(y, X\U Exercise 6. [Topology] Let X be a nonempty set and 7 = {U C; U = 0 or X\U is countable}. Prove that is a topology on X.
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