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.

For the topology consisting of the empty set, the whole real line, and upper open rays, a nonempty subset is compact exactly when it has a smallest element. An open set containing the smallest element contains the entire subset. Conversely, if there is no smallest element, the rays $(a,\infty )$ indexed by points $a$ of the subset cover it, but any finite selection misses its smallest chosen endpoint.