Exercise 1.12.b

Exercise 1.12.b: Let $$d$$ be any metric on any set $$X$$. Find an example to show that the intersection of infinitely many open sets may not be open for this arbitrary metric.

Olomana
In 1.12, we showed that the intersection of two open sets is open by taking the minimum of two epsilons. This works for finite intersections because a finite set of epsilons has a minimum. However, an infinite set of epsilons may or may not have a minimum.

Consider $$\mathbb{R}$$ under the standard metric and an infinite set U of balls about the origin:

$$U_i = B(0, \frac{1}{i})$$

The balls are all open, and our argument breaks down because there's no minimum epsilon. The intersection is the singleton set $$\{0\}$$, which is not open.

This gives us an example for a particular metric space.

Now consider $$\mathbb{R}$$ under the discrete metric. All subsets of $$\mathbb{R}$$ are open (just take $$\epsilon = \frac{1}{2}$$). Trivially, all infinite intersections of open sets are open in this metric space.

Olomana 06:39, March 15, 2011 (UTC)

'''Great, and thanks for clarifying that the arbitrary intersection of open sets may be open for some specific metrics. --Steven.clontz 18:24, March 16, 2011 (UTC)'''