HPAS Maths Optional PYQs: Metric Spaces

On that page, you will find year-wise and question-wise solutions.

This section covers the definitions, fundamental properties, topological concepts (open, closed, compact, connected), and completeness within metric spaces.

HPAS 2024 Metric Spaces Questions

Question 3(b): Show that \((R_{\infty}, d)\) defined using \(d(x,y)=|f(x)–f(y)|\) (where \(f(x)=x/(1+|x|)\)) is a bounded metric space.

Question 4(a): Let \((X,d)\) be a complete metric space and Y be a subspace of X. Show that Y is complete if and only if it is closed in \((X,d)\).

HPAS 2023 Metric Spaces Questions

Question 1(e): Show that every closed sphere is a closed set.

Question 4(b): Show that every compact subset F of a metric space \((X,d)\) is closed.

Question 4(c): Show that any disjoint pair of closed sets in X can be separated by disjoint open sets in X.

HPAS 2021 Metric Spaces Questions

Question 1(a): Show that every closed subspace of a complete metric space \((X,d)\) is complete.

Question 3(a): Show that the metric space \((X,d)\) defined by \(d(x,y)=\int_{0}^{1} |x(t)–y(t)| dt\) (on continuous functions) is not complete.

HPAS 2020 Metric Spaces Questions

Question 4(c): Define a connected metric space. Prove that every closed interval of the real line R is connected.

Question 5(a): Define a continuous function between metric spaces. Prove that any real-valued continuous function on a compact metric space is bounded.

HPAS 2019 Metric Spaces Questions

Question 4(a): Define a compact metric space. Prove that a closed subset of a compact space is compact.

HPAS 2018 Metric Spaces Questions

Question 1(b): Let \((X,d)\) be a metric space and \(A \subseteq X\). Then prove that \(\bar{A}=\{x \in X: d(x,A)=0\}\).

Question 6(a): Define an open sphere in a metric space. Describe the open sphere of unit radius centered at (0,0) for the Euclidean metric.

Question 6(b): Define a closed sphere in a metric space. Describe the closed sphere of unit radius centered at (0,0) for the metric \(d(z_1, z_2) = |x_1−x_2| + |y_1−y_2|\).

HPAS 2017 Metric Spaces Questions

Question 4(a): If \((A, d)\) is a metric space, then show that: \(|d(x_1,y_1)–d(x_2,y_2)| \le d(x_1,x_2)+d(y_1,y_2)\).

HPAS 2016 Metric Spaces Questions

Question 6(b): Show that the function \(d^*\) defined by: \(d^*(x,y)=\frac{d(x,y)}{1+d(x,y)}\) for all \(x,y \in X\) is a metric on X.

HPAS 2015 Metric Spaces Questions

Question 1(c): Prove that in a metric space every open sphere is an open set.

Question 4(b): Let \(f\) and \(g\) be complex continuous functions on a metric space \((A, d)\) then \(f+g\), \(fg\) and \(af\) are continuous on \((A, d)\).

HPAS 2014 Metric Spaces Questions

Question 1(d): Prove that the set of real numbers R and the function \(d(x,y)=|x−y|\) form a metric space.

Question 4(b): Prove that the metric space \((R, d)\) is complete, where \(d\) is the usual metric.