A topological subspace question about sets

Question

I am self studying topology. This is not homework

I am using Topology For Beginners by Steve Warner

I am on Chapter 9 I am having trouble understanding an example

Let T be standard topology on $\mathbb{R}$

Consider ${0}$ $\cup$ (1,2] as subspace of $\mathbb{R}$. $\{0\}$ is open in the subspace topology because $(-1,1) \cap (\{0\} \cup (1,2])= \{0\}$ is open in the subspace topology.

I don’t understand how $(-1,1)$ is gotten.

He goes onto to state $(1,2]$ is open in subspace topology since $(1,3) \cap (\{0\} \cup (1,2])=[1,2)$

The same is for $(1,3)$

Lastly, he states $\{0\}$ and $(1,2]$ are complements? The compliment of $(1,2]$ is still would be a half interval containing $(1,2]$

Complement of $\{0\}$ = $\mathbb{R}$ I can’t see it. Help

Answer 1

1. I don't understand how $(-1, 1)$ is gotten.

We use $(-1, 1)$ because it is an open subset of $\mathbb{R}$ and because $(-1, 1) \cap (\{0\} \cup (1, 2]) = \{0\}$. Recall that a set $S \subseteq \{0\} \cup (1, 2]$ is said to be open under the subset topology iff there is some open set $U \subseteq R$ such that $U \cap (\{0\} \cup (1, 2]) = S$. This is how we prove that $\{0\}$ is open in $\{0\} \cup (1, 2]$.

2. Lastly, he states $\{0\}$ and $(1, 2]$ are complements. I can't see it.

They are complements as subsets of $\{0\} \cup (1, 2]$.