I have a worksheet for my real analysis course that sets up a problem by defining "a new topology $S$ on $\mathbb R$ by declaring a set $U \subseteq \mathbb R$ to be in $S$ if $\mathbb R \setminus U$ is finite." We haven't covered topologies in class yet, but the worksheet says that one of the conditions for a thing $T$ to be a topology is that $\emptyset \in T$.

It seems to me that $\mathbb R \setminus \emptyset$ would not be finite, because it would simply be $\mathbb R$ which is not finite. So $\emptyset \subset S$, but $\emptyset \notin S$. This would mean $S$ is not a topology. Have I misunderstood something about topologies?

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    $\begingroup$ The cofinite topology is defined to contain all cofinite subsets, as well as the empty set. $\endgroup$ Oct 16 '19 at 23:22
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    $\begingroup$ You are correct, you have to explicitly add the empty set to $S$ to make it a topology. This is the so-called “co-finite” topology. $\endgroup$ Oct 16 '19 at 23:22
  • $\begingroup$ So as literally written, S would not be a topology? If so, is it a convention to implicitly add the empty set as needed when defining a topology? $\endgroup$ Oct 16 '19 at 23:24
  • $\begingroup$ Now that I've got a name, it looks like math.stackexchange.com/q/1876571/698711 is related. $\endgroup$ Oct 16 '19 at 23:28

You're correct, $\mathbb{R} \setminus \emptyset = \mathbb{R}$ which is not finite. To be a topology the empty set must be included and $S$ doesn't include the empty set. You seem to understand it correctly.

  • $\begingroup$ Thanks! Is it a convention to implicitly add the empty set as needed when defining a topology? $\endgroup$ Oct 16 '19 at 23:41
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    $\begingroup$ I would expect the empty set and the entire set to be tacitly included in the topology if they're claiming it is one. $\endgroup$ Oct 16 '19 at 23:44
  • $\begingroup$ @CyclotomicField: "tacitly" means the same as "implicitly". Did you mean "explicitly"? $\endgroup$
    – TonyK
    Oct 17 '19 at 0:33
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    $\begingroup$ For something to be explicit, it has to be stated right there for you to read. $\endgroup$
    – Lubin
    Oct 17 '19 at 0:54
  • $\begingroup$ @TonyK I meant tacitly. I'm just ridiculously pedantic and prefer rephrasing thing to avoid repetition. I don't know why I'm this way but I've come to accept it about myself. $\endgroup$ Oct 17 '19 at 1:41

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