Can a region of the $xy$ plane be infinite in area and topologically closed?

Question

Is it possible to create a region in the $xy$ plane that has infinite area but contains all of its boundary points? In other words, if I am introducing a theorem that begins: "Let R be a closed region in the $xy$ plane", do I need to also specify that R is finite?

Answer 1

To address your concern in the comments, it's a theorem that a topological space contains nontrivial clopen sets (sets other than $\emptyset$ and the space itself) $\iff$ the space is disconnected. Because $\mathbb{R}^2$ is connected, any nonempty, proper, closed subset of $\mathbb{R}^2$ is not also open.

Since closed sets are defined to be the complements of open sets, your question will have an affirmative answer if we can establish the existence of open sets of finite area. Such open sets exist; in particular, the metric topology has a basis consisting of the collection of open balls of finite radius.

Answer 2

What about the region $$ D = \left\{(x,y)\mid x \geq 1, 0 \leq y \leq 1/x\right\} $$ A few more examples came to mind, but then I wasn't sure on the definition of region.

Answer 3

If you want a proper subset: $\{(x,y)\mid 0\leq x \}$