Upper-ray topology on the real line ($\{\varnothing ,\mathbb{R}\}\cup \{(a,\infty ):a\in \mathbb{R}\}$)

These sets form a topology: a nonempty union of upper rays is another upper ray, or all of $\mathbb{R}$ when their endpoints have no lower bound; a finite intersection of rays has endpoint the maximum of their endpoints. The whole line must be included. The continuous map $n\mapsto n$ from $\mathbb{N}$ with the cofinite topology to this space is nonconstant, because each upper ray has cofinite inverse image.