Consider a $X, Y \subset \mathbb{R}$. We know that there are exist continuous, bijective $f : X \to Y$ and $g: Y \to X$. I found an example of non-homeomorphic and disconnected $X$ and $Y$ ( collection of open intervals and points , and collections of intervals and semi interval). But my teacher told that there is example of connected $X$ and $Y$ according to this properties. Also he said that it's possible to upgrade a standard counterexample with disjoint intervals. Any ideas ?
I thought to consider intervals $(a, + \inf)$ instead of $(i,i+1)$ and upgrade topology on $\mathbb{R}$. But don't know how to construct it.
There are no such spaces. A connected subspace of the real line is an interval or a point. Suppose $X,Y \subset \mathbb{R}$ are connected subspaces. If $X$ is compact and $f:X \to Y$ is a bijection, then it is a homeomorphism (since $Y$ is Hausdorff). Half-open intervals are homeomorphic. Open intervals are homeomorphic. So, the only remaining possibility (up to reordering and homeomorphism) is that $X = [0,1)$ and $Y = (0,1)$. But a bijection $f:X \to Y$ must have the property that either $(0,f(0))$ is not in the image or $(f(0),1)$ is not in the image.