Metric Space | Most Repeated Questions |Previous Year Questions with Solution |PU & Other University

How will you define a metric Space .
Let R be the real line,define a function d by d(x,y) = |x-y| for all x,y€R .Show that d is a metric on R and (R,d) is a metric Space.

Let (X,d) be a metric Space. Define a function p by p(x,y) = min{1,d(x,y)} for all x,y€X .Show that p is also a metric on X.

Let (X,d) be a metric Space and G is a sunset of X .Show that G is an open set in X if and only if G is union of open spheres.

In a metric Space (X,d) .Show that arbitrary union of open sets is also an open set.

Define Convergent Sequence in a metric Space (X,d) .Show that every convergent sequence in (X,d) is a Cauchy Sequence.

When a metric Space (X,d) complete ? Show that the real line R w.r.t usual metric is complete ?

Explain any two of the following in a metric Space.
(a) Neighborhood of a point
(b) Limit point or accumulation point
(c) Contraction mapping

Define a closed set in a metric Space (X,d). Show that a finite union of closed sets in (X,d) is also a closed set.

Theorem:
State and Prove Characterization of continuity in a metric Space in terms of open sets.

Metric Space || Previous Year Questions with Solution || Important for all University Exams
Previous year Questions of Patna University & others.
Most Repeated Questions on Metric Space

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