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Say we have an infinite set $X$ and a topology $T$. In the general defition of a topology the empty set $\emptyset$ and the whole set $X$ belong to the topology. But isn't that contradictory with the defintion of the cofinite topology.

Cofinite Topology:

$T = \{ A \subset X | A = \emptyset $ or $ X\setminus A$ is finite$\}$

So according to the definition of a topology we have that $X,\emptyset \in T$ and according to the cofinite topology we only have $\emptyset \in T$ (The whole set $X$ is closed).

Gr

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5 Answers 5

up vote 3 down vote accepted

No it is not: Let $A=X\subset X$, then $X \backslash X = \emptyset$ which is clearly finite, indeed $|\emptyset|=0$, i.e. the cardinality of the empty set is zero. Hence $X\in T$

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So, why is it then, that the empty set is used in the definition. Isn't that trivial, regarding the fact that every topology has the whole set and the empty set as one of its members? –  omar Jan 10 '13 at 13:49
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@aydogan You have a general definition of a topology. Now you can write down the definition of the cofinite topology and you have to prove that it is really a topology. Without "adding" the empty set, it would not guarantee that the empty set is open. As soon as $X$ is not finite, you have that $X\backslash \emptyset = X$, which is in this case not finite. –  user8 Jan 10 '13 at 13:52

The cofinite topology is the smallest topology in which all the cofinite sets are open.

As it turns out this is just the cofinite sets, and the empty set added. Because people often hate carrying around extra words, like "generated by" or "smallest topology containing", in this case where the only set missing is the empty set itself - we abuse the language and just say the cofinite topology.

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$X\in T$ as $X \setminus X = \emptyset$ which is finite.

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The whole set $X$ is closed indeed, but that does not mean it cannot be open. In topology close does not imply not open and vice (not) versa. And empty set and whole space $X$ are always clopen (both close and open) sets in any topology.

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In case, when X is finite,X\empty set=X is finite,but when X is infinite,then X\empty set=X is infinite and is not open obviously.To remove this ambiguity we must mention the inclusion of empty set, separately, in the collection of all co finite subset of X thus it as ,tau is the collection containing empty set and all cofinite subset of X. i.e tua ={A:A is any cofinite subset of X}U{{}} This becomes cofinite topology on X.

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