Closed sets in a cofinite topology

Question

I'm reading a set theory text book and I've come upon the section on topologies. (An Introduction to Set Theory and Topology by Ronald Freiwald, p. 105). The author defines the cofinite topology in the usual way:

T = {O ⊆ X : O = ∅ or X - O is finite}

Then he writes:

"In (X,T), a set F is closed iff F = ∅ or F is finite."

This seems wrong to me. What if X is infinite? Then it is closed but not finite. Am I missing something? If this is wrong, what do you think he was trying to get at?

Did he mean $F$ is closed iff $F=\color{blue}X$ or $F$ is finite? — J. W. Tanner, Apr 28 '21 at 00:14
That's definitely an error in the text. It should read $F = X$ or $F$ is finite. From the spirit of the introduction to the book, I am sure the author would appreciate it if you let him know about the error. — Rob Arthan, Apr 28 '21 at 00:19

score 10 · Accepted Answer · answered Apr 28 '21 at 00:23

10

I think you found a typographical error.

I think the author meant that, in the cofinite topology, a set $F$ is closed iff $F=\color{red}X$ or $F$ is finite.

answered Apr 28 '21 at 00:23

J. W. Tanner

60,406

score 1 · Answer 2 · edited Apr 30 '21 at 01:38

1

$$\begin{align*} F\text{ is closed}&\iff X-F\in T\\ &\iff X-F=\emptyset\text{ or }X-(X-F)\text{ is finite}\\ &\iff F=X\text{ or }F\text{ is finite}. \end{align*}$$

edited Apr 30 '21 at 01:38

Arturo Magidin

398,050

answered Apr 28 '21 at 00:28

hamam_Abdallah

62,951

Did you mean $F=X$ or $\color{red}F$ is finite? – J. W. Tanner Apr 28 '21 at 00:36
1

@J.W.Tanner Yes but with the blue color. – hamam_Abdallah Apr 28 '21 at 01:19
You could align it better using display math and align... – Arturo Magidin Apr 28 '21 at 01:24

Closed sets in a cofinite topology

2 Answers2