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I'm looking to gain a better understanding of how the cofinite topology applies to R. I know the definition for this topology but I'm specifically looking to find some properties such as the closure, interior, set of limit points, or the boundary set and how these change based on whether a subset A in R is closed, open, or clopen.

Any help would be appreciated.

Note: I have only the most basic of definitions for closure, interior, etc.

Thank you!

Asaf Karagila
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John MatthewS
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3 Answers3

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In the cofinite topology, '$A$ is closed' means '$A$ is finite or is $\mathbb{R}$' and '$A$ is open' means '$A$ has finite complement or is empty'. In particular...

  • If $A \subseteq \mathbb{R}$ is infinite then the only closed set containing $A$ is $\mathbb{R}$, and hence $\operatorname{cl} A = \mathbb{R}$. In particular, if $A$ is infinite, every point of $\mathbb{R} - A$ is a limit point of $A$. Hence if $A$ is infinite then $\partial A = \mathbb{R} - A$.
  • If $A \subseteq \mathbb{R}$ has infinite compliment then its only open subset is $\varnothing$, so $\operatorname{int} A = \varnothing$.
  • The only clopen sets are $\mathbb{R}$ and $\varnothing$.
Clive Newstead
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The co-finite topology has a good property: It's the minimal $T_1$ topology, so every other property (Hausdorff, Regular, etc) you might expect to fail.

For example, every closed set is either finite or $\mathbb{R}$. So, the closure of any set is the set iself (if it's finite) or $\mathbb{R}$ if it is infinite.

Every open set is infinite or the empty set, so the interior of any finite set is $\emptyset$.

So there are no clopen sets (except $\mathbb{R}$ and $\emptyset$)

The boundary $\partial A = \overline{A} \cap \overline{\mathbb{R}\setminus A}$, so, if $A$ is open, its boundary is precisely $\mathbb{R}\setminus A$. If $A$ is closed, the closure of $\mathbb{R}\setminus A$ is $\mathbb{R}$ and then $\partial A = \overline{A} = A$

The set of limit points of $A$ is $\emptyset$ if $A$ is closed, and is the set itself if it's open.

Jonas Gomes
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So to answer your first question if $O$ is an open set that isn't empty in the co-finite topology then by definition $\bar{O}$ is the smallest closed set that contains $O$. Now we know that $O$ has infinitly many elements and the only closed set that doesn't only have finitely many elements is $\mathbb{R}$ so it must be $\bar{O}$.

The boundary of an open set $O$ is $\bar{O} \backslash O$ so if $O$ isn't empty then in this topology its boundary is $\mathbb{R} \backslash O$.

  • How bout for closed sets? Also, can't an infinite set have an infinite compliment? Like (-infinities,0) in R? – John MatthewS Oct 31 '14 at 00:23
  • @JohnMatthewS I don't think I see how your second question connects to this? –  Oct 31 '14 at 01:12
  • Closed sets (that aren't the whole space) are finite and are their own closure. Their interior is the biggest open set inside of them which has to be the empty set since everything else that is open is infinite. Their boundary is their closure minus their interior which is themselves again. –  Oct 31 '14 at 01:13
  • Isn't (0,infinity) closed because it's compliment is not finite? – John MatthewS Oct 31 '14 at 02:43
  • @JohnMatthewS no a set is closed if its compliment is open. And in this topology a set is open if its compliment is finite (or the whole space). Also the compliment of the compliment of a set is just the set again so the only closed sets (that aren't $\mathbb{R}$) are finite. $(0,\infty)$ is neither open nor closed in this topology. –  Oct 31 '14 at 02:47