Prove that a closed interval $A=[0,1]$ and $B=\{(x,y)∈R^2 \mid ||(x,y)||≤1\}$ are not manifold
I'm struck with this problem.Can anyone explain how and what property should i use to show that for any open sets in A and B which contains boundary point is not homeomorphic to open set in $R$ and $R^2$ respectively
Consider an open neighbourhood $U$ of $x$ in $\mathbb R$. For simplicity assume $U$ is connected (i.e. replace it with a possibly smaller open ball). Then $U\setminus\{x\}$ is not connected. This property does not hold if $x$ is instead a boundary point of a closed interval.
A similar argument works in $\mathbb R^2$: For an interior point there exists a closed line $S^1\subset U\ni x$ that can be contracted in $U$, but not in $U\setminus\{x\}$. For a boundary point, this is not the case.