## Posts (1) containing the 'math.An’ (Analysis) tag:## Counterexamples 1/N |9 January 2023|
As nicely felt through the 1964 book This short post highlights a few of my favourite examples from the book by Gelbaum and Olmsted. *A function that is everywhere finite and everywhere locally unbounded.*
A function is locally bounded when for each point in its domain one can find a neighbourhood such that is bounded on . Now consider the function defined by Now assume that we could locally bound on (some open neighbourhood in ), that means that is bounded on such a neighbourhood and hence, so is . However, that would imply that only contains finitely many rational numbers! We plot (approximately) below over for just points and observe precisely what one would expect. *Two disjoint closed sets that are at a zero distance.*
This example is not that surprising, but it nicely illustrates the pitfall of thinking of sets as ‘‘ *An open and closed metric ball with the same center and radius such that the closure of the open ball is unequal to the closed ball.*
This last example is again not that surprising if you are used to go beyond Euclidean spaces. Let be a metric space under the metric Now assume that is not simply a singleton. Then, let be the open metric ball of radius centered at and similarly, let be the closed metric ball of radius centered at . Indeed, whereas . Now to construct the closure of , we must first understand the topology induced by the metric . In other words, we must understand the limit points of . Indeed, is the Further examples I particularly enjoyed are |