TIFR Mathematics 2024 | Part B | Question: 2

Question

There exists a metric space $\text{X}$ such that the number of open subsets of $\text{X}$ is exactly $2024$.

Answer 1

Let's consider a discrete metric space with a finite number of points. In a discrete metric space, the metric between distinct points is 1, and the metric of any point with itself is 0.

Now, let X be a set with 2024 distinct points. The discrete metric space on X is defined as follows:

• For any two distinct points x and y in X, the distance d(x, y) is 1.
• For any point x in X, the distance d(x, x) is 0.

In this metric space, every singleton set {x} is an open subset because for any point x in X, you can choose an open ball of radius 0.5 centered at x, and it will contain only x itself.

Additionally, the union of any collection of open subsets is open. Since there are 2024 distinct points in X, there are 2^2024 possible subsets (including the empty set and the entire set). However, not all of them will be open. The open subsets in this case are precisely the singleton sets, which means there are 2024 open subsets.

Therefore, you can construct a metric space X with exactly 2024 open subsets by considering a discrete metric space on a set with 2024 distinct points.