Questions on "simple-connectedness-like" property

Question

I wanted to know if there's any notion which is very similar to the simple connectedness, but defined "purely" in terms of points and sets.

Here's my attempt to do it. Let $X$ be a topological space. If there exists open connected subsets $U$, $V$ in $X$ such that $U \cup V = X$ but $U \cap V$ is disconnected, let's say that $X$ is non-simply onnected. (It doesn't agree with the usual definition. But for simplicity, let's call it just simple connectedness) If $X$ is not non-simply connected, let's say that $X$ is simply connected.

I want to know if the new notion of simple connectedness satisfies the followings: (but it was difficult for me)

1. If $X$ is simply connected and $r:X \to A$ is a retraction, $A$ is also simply connected.
2. Let $U, V$ be open subsets of $X$ such that $U \cup V = X$. If $U, V$ are simply connected and $U \cap V$ is connected, then $X$ is simply connected.
3. If $X$ and $Y$ are simply connected, $X\times Y$ is also simply conncected.

Can anyone prove/disprove these statements?

Answer 1

Your property is called "open-unicoherence"; see this article, "A survey on unicoherence and related properties", by A. García-Máynez and Alejandro Illanes.