# general topology – Why is my test that \$ mathbb R \$ disconnected incorrect?

The definition of connectivity in my notes is:
A topological space $$X$$ is connected if a pair of non-empty subsets does not exist $$U$$, $$V$$ such that $$U cap V = emptyset$$ Y $$U cup V = X$$.

However, if I have the subsets $$(- infty, 0]$$ Y $$(0, infty)$$ then these are disjoint and cover $$mathbb R$$ and therefore $$mathbb R$$ is disconnected

but nevertheless $$mathbb R$$ It is clearly connected. Where have I been wrong?