Attempt: Subspace of $R$ by removing a point has nontrivial clopen set, but Subspace of $R^n$ by removing a point has no clopen set. Since has clopen set is a topological invariant, the two subspace is not homeomorphism. So the original spaces are not homeomorphic.
But I cannot figure out why subspace of $R^n$ by removing a point has no nontrivial clopen set. Any help? Thanks.
The question not allows to specially use the definition of connected. Or it only allows to use that having no nontrivial clopen set is topological invariant. (Although having no nontrivial clopen set implies connected...)
First, your question is somewhat misleading:
The condition that you're asking for is equivalent to connectedness. In particular, a space $X$ is disconnected if there exist nonempty open sets $U$ and $V$ so that $X=U\cup V$ and $U\cap V=\emptyset$. Moreover, if $W$ is a nontrivial clopen set, then $W$ and $W^c$ are both open, $X=W\cup W^c$, and $W\cap W^c=\emptyset$. Therefore, they form a separation of $X$. Hence, by asking for a nontrivial clopen set, you are asking for a separation and using the definition of disconnected.
What I think that you are asking is to avoid the implication path connected implies connected. So, to prove that $\mathbb{R}^n$ is connected without going through path connected.
Suppose that $\mathbb{R}^n\setminus\{0\}$ has a nontrivial clopen set $X$. Then, since $\mathbb{R}^n\setminus\{0\}$ has the induced topology, there is an open set $U$ in $\mathbb{R}^n$ and a closed set $D$ in $\mathbb{R}^n$ so that $U\cap \mathbb{R}^n\setminus\{0\}=X$ and $D\cap\mathbb{R}^n\setminus\{0\}=X$. We note that $U$ and $D$ only differ by a single point $\{0\}$ since their intersection with $\mathbb{R}^n\setminus\{0\}$ is the same. Moreover, the two sets must differ because $\mathbb{R}^n$ is connected and has non nontrivial clopen sets.
Let $x\in X$ and let $y\not\in X$. By perturbing $x$ and $y$ slightly, we may assume that the line $\ell$ though $x$ and $y$ does not include $0$ (since $X$ is clopen, there are small neighborhoods around $x$ in $X$ and around $y$ in in $X^c$). Then the induced topology from $\mathbb{R}^n$ to $\ell$ produces the standard topology on $\ell\simeq\mathbb{R}$. We observe that $U\cap \ell=D\cap \ell$, since their only difference is at $0$. Since $x\in U\cap\ell$ and $y\not\in U\cap \ell$, this defines a nontrivial clopen set in $\mathbb{R}$, which contradicts the fact that $\mathbb{R}$ is connected.