Consider $X = \{ f :\Bbb R^n \to \Bbb R^n \mid f \text{ continuous bijection } \} $ endowed with the product topology inherited possibly from $\Bbb R^{{n}^{\Bbb R^n}}$ (terrible notation). Is $X$ connected or disconnected?
When $n = 1$ I think I've managed to figure out that $X$ can be partitioned to $A = \{f \in X \mid f \text{ increasing }\}$ and $B = \{f \in X \mid f \text{ decreasing } \}$ and so $$X = A \cup B$$ and $A \cap B = \emptyset$ making $X$ disconnected.
The problem with $n > 1$ is that we only have this sort of characterization of bijective functions to increasing and decreasing sets in $\Bbb R$ as there is some kind of order defined in it. We lose this when we go up dimensions. Is there some "invariant" or another clever way to classify bijections in larger dimensions of $\Bbb R$ that we could use to find a separation for $X$? Or is it even true that $X$ is disconnected when $n > 1$?