All Questions

Let $X \subseteq \mathbb{R}^n$, and assume there is a strong deformation retract from $\mathbb{R}^n$ to $X$. Is $X$ necessarily simply connected at infinity? (Edit) Follow up question: if there is a ...

Let $X$ be a compact $T_0$ topological space such that its closed and connected subsets are path connected. Is there any characterization for such a space?

Recall a topological space $X$ is path connected if for all $x,y \in X$ there is a continuous function $f\colon [0,1] \to X$ such that $f(0)=x$ and $f(1)=y$. Say that $X$ is continuously path ...