A set is closed when what is left of the space outside it is open. Nothing new is being assumed: every fact about closed sets is a fact about open ones, read through the complement.

Describing a shape by the hole it leaves in a sheet of paper. Cut the hole and the shape is fixed, so anything true of one is true of the other, turned round.

A subset $C$ of $X$ is closed when its complement $X\setminus C$ is open. Taking complements turns a union into an intersection and back, so the two rules of the axioms swap over: closed sets may be intersected in any number and united only finitely many at a time.

• Closed does not mean not open. $\varnothing$ and $X$ are both open and closed in every topology, since each is the other’s complement.
• On the real line $(0,1)$ is open, $[0,1]$ is closed, and $$ is neither.

$C$ is closed exactly when what is left of $X$ outside it is open. Nothing new is assumed by the word: every fact about closed sets is a fact about open ones, read through the complement.

Work with complements. A point lies outside an intersection exactly when it lies outside at least one of the members, which is what the union of the complements says. Each of those complements is open, because each member is closed. A union of open sets is open with no restriction on how many, so the complement of the intersection is open, and that is what being closed means. The size of the collection never enters, which is why it may be infinite.

A point misses the intersection exactly when it misses at least one member, and that is what the union of the complements collects.

Each member is closed, so by the definition each of these complements is open.

A union of open sets is open, and the axiom puts no limit on how many there are.

Its complement has just been shown open, which is the definition of closed.

A union of two open intervals, so it is open, so $[0,1]$ is closed.

Not open, because no interval around 0 fits inside; not closed, because its complement is not open either.

Missing both is the same as missing their union.

Every member closed, the union not closed.

Read every question about closed sets through the complement; the topology only ever lists open sets.