My general topology textbook introduced T$_1$-spaces as the following:
A topological space $(X,\tau)$ is said to be a T$_1$-space if:
- $\forall x \in X$, the set $\{x\}$ is closed
They also stated the following proposition:
If $(X,\tau)$ is a topological space, then:
(1) $X$ and $\emptyset$ are closed sets
(2) The intersection of any (finite or infinite) number of closed sets is closed
(3) The union of a finite number of close sets is also closed
Now, let $A \subseteq X$. We have that $A= \bigcup_{x \in A} \{x\}$. If $A$ is finite, then $A$ is the union of a finite number of sets (3), thus every finite subset is closed. If $A$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed. Doesn't this make $\tau$ the finite-closed topology?
Best Answer
The mistake is at
"If $A$ is infinite, then this is the union of a infinite number of closed sets, hence it's not closed"
In general, $p\to q$ is not the same as $\sim p\to \sim q$.
There are $T_1$ space which does not have finite-closed topology. $\mathbb R$ with the standard topology is one of the example.