Bases and subbases of a topology

Let $X$ be a set, and $\beta_{0}$ an arbitrary family of its subsets forming a cover of $X$. Prove that there exists a unique topology on $X$ for which $\beta_{0}$ serves as a subbase.

Prove that a system of open subsets $\beta$ of a topological space $X$ forms a base of the topology $\tau$ if and only if every element of $\tau$ is a union of some elements of $\beta$.

Let $\tau_{1}$ and $\tau_{2}$ be two topologies on a set $X$ with bases $\beta_{1}$ and $\beta_{2}$ respectively. Prove that $\tau_{1}$ is weaker than $\tau_{2}$ if and only if for every $B_{1} \in \beta_{1}$ and every point $x \in B_{1}$ there exists $B_{2} \in \beta_{2}$ such that $x \in B_{2} \subset B_{1}$.

Prove the stated assertion. If $\mathcal{D}$ is a base of the topology $\Omega$ of a topological space $X$ and $\mathcal{D} \subset \mathcal{D}_{1} \subset \Omega$, then $\mathcal{D}_{1}$ is also a base of the topology of this space.

Prove the stated assertion. The union of the local bases of the topology of all points of a topological space forms a base of the topology of this space.

Show that the trivial topology on a set has a unique base.

Regarding Proposition 2, show that if $\mathcal{B}$ is a base for a topology, then properties (i) and (ii) hold.

Proposition 2: For a non-empty set $X$, let $\mathcal{B}$ be a collection of subsets of $X$. Then $\mathcal{B}$ is a base for a topology for $X$ if and only if (i) $\mathcal{B}$ covers $X$, that is, $X = \bigcup_{B \in \mathcal{B}} B$; and (ii) if $B_{1}$ and $B_{2}$ are in $\mathcal{B}$ and $x \in B_{1} \cap B_{2}$, then there is a set $B$ in $\mathcal{B}$ for which $x \in B \subseteq B_{1} \cap B_{2}$.

Let $\mathcal{T}_{1}$ and $\mathcal{T}_{2}$ be topologies for a non-empty set $X$. Show that $\mathcal{T}_{1} = \mathcal{T}_{2}$ if and only if there are bases $\mathcal{B}_{1}$ for $\mathcal{T}_{1}$ and $\mathcal{B}_{2}$ for $\mathcal{T}_{2}$ that are related as follows at each point $x$ in $X$ : for each neighborhood $\mathcal{N}_{1}$ of $x$ belonging to $\mathcal{B}_{1}$, there is a neighborhood $\mathcal{N}_{2}$ of $x$ belonging to $\mathcal{B}_{2}$ for which $\mathcal{N}_{2} \subseteq \mathcal{N}_{1}$ and for each neighborhood $\mathcal{N}_{2}$ of $x$ belonging to $\mathcal{B}_{2}$, there is a neighborhood $\mathcal{N}_{1}$ of $x$ belonging to $\mathcal{B}_{1}$ for which $\mathcal{N}_{1} \subseteq \mathcal{N}_{2}$.

A collection $\mathcal{B} = \left\{ B_i : i \in I\right\}$ of subsets of $X$ is called a basis and its elements are called basis elements if the following properties are satisfied:

Show that the set of balls $\left\{ B(x, r), x \in X, r > 0\right\}$ is a basis for the topology of a metric space $X$; that is, show that it is a basis and the topology it determines is the metric topology.

Show that any open set in $T_{\mathcal{B}}$ (the topology determined by a basis $\mathcal{B}$) is a union of basis elements.

Show that the open intervals give a basis for the topology of $\mathbb {R}$.