How can the empty set be open in the definition of the cofinite topology?

Question

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

Answer 1

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 $X\in T$

Answer 2

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.

Answer 3

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

Answer 4

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.