Proof that function is a homeomorphism

Question

Let $ (Y \times Y, \tau )$ be a connected topological space and $ P(Y) $ be a subset of $Y \times Y$ defined by $P(Y) = \{(a,b)\in Y \times Y:a\not =b \} $. Let $ \Lambda:P(Y) \rightarrow P(Y) $ be a function defined by $ (a,b) \mapsto (b,a)$.

How do you prove that $ \Lambda $ is a homeomorphism from $ P(Y) $ on to itself?

Answer 1

Since $\Lambda$ is bijective (because $\Lambda^{-1} = \Lambda$), it suffices to check that $\Lambda$ is continuous. Take $U \times V$ an open neighbourhood of $(b,a) = \Lambda(a,b)$. Then $V \times U$ is an open neighbourhood of $(a,b)$. Surely $\Lambda(V \times U)\subseteq U \times V$ (actually equal), and we're done.