in Section 1.3 "Connected and compact sets" from the book "Lecture notes on elementary Topology and Geometry" by Singer & Thorpe, Theorem 2 states the following:
Theorem: Let S be a topological space (with U as the topology), and let $T_{0}$ and {$T_{w}$} with ${w \in W}$ connected subsets of S (that is, connected with the relative topology or subspace topology). Assume $T_{0} \cap T_{w} \neq \varnothing$ for each ${w \in W}$. Then, $T_{0} \cup (\cup_{w\in W} T_{w})$ is connected.
This can be used to prove that $\mathbb{R^n}$ is connected, because $\mathbb{R^n}$= {0} $\cup (\cup_{w\in W} T_{w})$ where $T_{w}$ are all the lines through $0$ and $W$ can be the unit sphere in $\mathbb{R^n}$.
So I have two questions.
- If $n=2$ how can I use this theorem to prove that $\mathbb{R^2}$-{0} is connected?
- Can I use the answer to 1. to prove that $\mathbb{R^n}$-{0} is connected?
I ask firstly for $n=2$ because that is what I used to understand the proof that $\mathbb{R^n}$ is connected, but I really want to undestand the general case. Thanks in advance.
Let $T_0=\Bbb R\times \{1\}$ and let $A=\{T_w\}_{w\in W}$ be the set of all the lines in $\Bbb R^2\setminus \{(0,0)\}$ that intersect $T_0.$ Every $(x,y)\in \Bbb R^2\setminus \{0\}$ belongs to a member of $A:$ If $x\ne 0$ then $(x,y)\in \{x\}\times \Bbb R\in A.$ If $y\ne 0$ the line $\{(u,u+y):u\in \Bbb R\}$ thru $(0,y)$ and $(1,1+y))$ belongs to $A.$
Similarly for $\Bbb R^n \setminus \{0\}^n$ when $n\geq 3.$ Every point in $\Bbb R^n\setminus \{0\}^n$ lies on a line in $\Bbb R^n\setminus \{0\}^n $that intersects the line $\Bbb R\times \{1\}^{n-1}.$