Topology axioms and open and closed sets

Let $(X, \tau )$ be a topological space, and $\sigma$ the collection of all its closed sets. Prove that

1. $\varnothing , X \in \sigma$;

2. the intersection of an arbitrary family of sets from $\sigma$ belongs to $\sigma$;

3. the union of any finite family of sets from $\sigma$ belongs to $\sigma$. Suppose, conversely, that $X$ is some set, and $\sigma$ a family of its subsets satisfying conditions (1), (2), (3). Prove that the collection of subsets $\tau$, consisting of the complements of sets from $\sigma$, is a topology on $X$.

Prove the stated assertion. The union of a finite collection and the intersection of any collection of closed sets is a closed set.

Determine whether the following subsets of the plane are open. Justify your answers.

Verify that the definition of an open set for a metric space satisfies the requirements for a topology.

Show that $[a, b]$ is a closed set in $\mathbb {R}$. Show that a rectangle (including the perimeter) is a closed set in $\mathbb {R}^{2}$.

Show that $[a, b)$ is neither open nor closed in $\mathbb {R}$.