S154 definition.

[Almanzoris]

I got confused after reading the description of this space. That previous way, the space would be discrete.

$\{\infty\}$ would be open. And if $x \in X \setminus \{\infty\}$, then $\{x\} = (X \setminus \{\infty\}) \cap (\{\infty, x\})$, which is open.

In the first edition of Counterexamples in Topology, it is defined that way. Is it updated in the second edition?

Nevermind, I had misread it. It clearly meant that the complement* included $\{infty\}$.

[prabau]

Like you said, the correct way to read this is that the complement contains $\infty$. The second edition has: "... declaring open any set whose complement either is finite or includes $p$". (a little clearer thanks to the word "either")
But I agree that this way to express things is still unnecessarily hard to figure out.

There are several spaces with a similar description:

• S20 (Fort space on a countably infinite set)
• S154 (Fort space on the real numbers)
• S24 (Modified Fort space on the real numbers)
• S22 (Fortissimo space on the real numbers)

I personally find the description of S24 easier to grasp. If we want to follow that for S154, it could be something like:

Let $X=\mathbb R\cup\{\infty\}$. Every point of $\mathbb R$ is isolated and the open neighborhoods of the point $\infty$ are the cofinite subsets of $X$ containing that point.
What do you think?

We could also mention in the text that this space is the one-point compactification of an uncountable discrete space (compare with S22 and S20).

[Almanzoris]

I agree with you. I have updated it now.