- #1

nightingale123

- 25

- 2

## Homework Statement

We define ##X=\mathbb{N}^2\cup\{(0,0)\}## and ##\tau## ( the family of open sets) like this

##U\in\tau\iff(0,0)\notin U\lor \exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}##

##a)## Show that ##\tau## satisfies that axioms for open sets

##b)## Show that ##(0,0)## lies in the Closure of ##\mathbb{N}^2##

##c)## Describe closed sets in topology ##\tau##

##d)## show that there doesn't exists a sequence##(x_{n})_{n\in\mathbb{N}}\subset\mathbb{N}^2 \text{ for which }\lim x_{n}\xrightarrow[n->\infty]{X}(0,0)##.

Assume that X is not first-countable

##e)##

## Homework Equations

## The Attempt at a Solution

I'm having trouble visualizing the sets in ##\tau## I know from the first part that the every point which is not ##(0,0)## is open. Also I know that every union of such sets will also be open. Therefore ##\mathbb{N}^2## in itself is open. However I don't know how to visualize the other condition ##\exists N\ni : n\in\mathbb{N},n>N\implies(\{n\}\times\mathbb{N})\backslash U\text{ is finite}## I would really appreciate it if someone could explain to me how this sets look as I am unable to continue with the problem.

Thank you