(x, 0) in the negative half space

Question

My course textbook writes this about weight spaces when talking about linear classification: .

Why would $w^{T}x$ be considered to be in the negative half space if $0$ is not less than $0$?

Edit: for further clarification, I'm confused as to the line " similarly, the set of weight vectors...this is also a half space" because it seems the case $(x,0)$ is classified in the negative half space because of the logic that $w^Tx<0$, which by the given label - $0$ - is not less than $0$.

Edit 2: Here are the slides talking about half spaces

Answer 1

Note that each weight vector $w$ defines a (linear) classifier $f_w$ by $$f_w(x) = \begin{cases} 1, & \text{if } w^T x \geq 0 \\ 0, & \text{otherwise, i.e. } w^T x < 0 \end{cases}$$ You are confusing $w^T x$ with $f_w(x)$, presumably because the book conflates the weight vector $w$ and the associated classifier $f_w$.