2. Let X be a non-empty set. Define the discrete metric on X and answer each of the following: (i) Identify the collection of open subsets in a discrete metric space. (ii) Show that a sequence in a discrete metric space converges if and only if it is eventually constant.
2. Let X be a non-empty set. Define the discrete metric on X and answer each of the following: (i) Identify the collection of open subsets in a discrete metric space. (ii) Show that a sequence in a discrete metric space converges if and only if it is eventually constant.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please solve number 2
Transcribed Image Text:2. Let X be a non-empty set. Define the discrete metric on X and
answer each of the following:
(i) Identify the collection of open subsets in a discrete metric space.
(ii) Show that a sequence in a discrete metric space converges if and
only if it is eventually constant.
3. Suppose that X and Y are two sets and f: X→ Y is a function.
Prove each of the following:
(i) If A and B are two subsets of X and AC B, then fƒ(A) c f(B).
(ii) If {Hala e A} is an indexed family of subsets of X, then
f(UaEA Ha) = Uae^f (Ha).
(iii) If {Gala E A} is an indexed family of subsets of Y, then
f-¹ (nae^Ga) nae^ f-¹(Ga).
=
3. Let (X, d) be a metric space. Prove each of the following:
(i) A closed ball is closed.
(ii) Intersection of two open sets is open
d(x,y)
(iii) If p(x, y) = 1+d(x,y)'
then p(x, y) is a metric on X.
4. Let XRx R, where R is the set of reals and let d be the Euclidean
metric on X. Show that the set {(x, y) = X\x² + y² < 1} is open.
Expert Solution
Step 1
First we have to define the discrete metric on .
Definition (Discrete Metric): The discrete metric on a non-empty set is defined as:
where .
The with above metric is called discrete metric space.
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