- #1

sa1988

- 222

- 23

## Homework Statement

Let ##U = \{(x,y)\in R^2 : xy > 0\}##

Show that ##U## is open in the product topology on ##R^2## induced from the standard topology on ##R##

## Homework Equations

## The Attempt at a Solution

Proofs are my downfall.

First I've visualised ##U## and it seems to be that it's essentially defined by the upper-right and lower-left quadrants of the x-y plane, not including the points on the axes themselves.

From what I understand, for the proof I need to show that every point ##t \in U## has an open neighbourhood, i.e the neighbourhood is entirely within ##U##.

I believe the best way would be to assume ##U## is closed, then look for a contradiction.

So, assuming ##U## to be closed,

its compliment ##U^c## must be open.

But ##U^c = \{ (x'y')\in R^2 : x'y' \leq 0 \}##

which has a clear limit at the point ##x = 0## or ##y = 0##

so there are neighbourhoods ##N## around the points ##n = (0,y')## and ##n = (x',0)## where ##n \in N## but ##n \notin U^c##

Hence ##U^c## must be closed, so ##U## is open.