Equivalence of Definitions of Topology
Contents
Theorem
The following definitions of the concept of Topology are equivalent:
Definition 1
Let $S$ be a set such that $S \ne \varnothing$.
A topology on $S$ is a subset $\tau \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ that satisfies the open set axioms:
\((O1)\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | ||||||
\((O2)\) | $:$ | The intersection of any two elements of $\tau$ is an element of $\tau$. | ||||||
\((O3)\) | $:$ | $S$ is an element of $\tau$. |
If $\tau$ is a topology on $S$, then $\left({S, \tau}\right)$ is called a topological space.
The elements of $\tau$ are called the open sets of $\left({S, \tau}\right)$.
Definition 2
Let $S$ be a set such that $S \ne \varnothing$.
A topology on $S$ is a subset $\tau \subseteq \mathcal P \left({S}\right)$ of the power set of $S$ that satisfies the following axioms:
\((O1')\) | $:$ | The union of an arbitrary subset of $\tau$ is an element of $\tau$. | ||||||
\((O2')\) | $:$ | The intersection of any finite subset of $\tau$ is an element of $\tau$. |
Proof
Definition 1 implies Definition 2
Let $\tau$ be a topology on $S$ by definition 1.
$O 1$ is the same as $O 1'$, so $O 1'$ holds for $\tau$.
From General Intersection Property of Topological Space:
- if $O 2$ holds, then $O 2'$ holds.
Thus $\tau$ is a topology on $S$ by definition 2.
$\Box$
Definition 2 implies Definition 1
Let $\tau$ be a topology on $S$ by definition 2.
$O 1'$ is the same as $O 1$, so $O 1$ holds for $\tau$.
$O 2'$ states that the intersection of any finite subset of $\tau$ is an element of $\tau$.
This applies when the subset of $\tau$ contains exactly $2$ sets.
Thus $O 2$ is a direct consequence of $O 2'$.
Also as a consequence of $O 2$, it follows that the intersection of an empty subset of $\tau$ is an element of $\tau$.
From Intersection of Empty Set it follows that $S \in \tau$.
So $O 3$ is a direct consequence of $O 2'$.
Thus all the open set axioms hold.
Thus $\tau$ is a topology on $S$ by definition 1.
$\blacksquare$