## Understanding Three “Standard” Topologies

Today we are going to be discussing three topologies that are seemingly “standard” among all others: the Order Topology, Product Topology, and Subspace Topology. They are easily predictable by their names and individual properties. We start with the Order Topology.

The Order Topology is a topology defined by some order relation like $<$ on the set $X$. The only subsets of $X$ are intervals given $a,b\in X$:

• Open interval
• $(a,b)$
• Half-open intervals
• $(a,b]$
• $[a,b)$
• Closed interval
• $[a,b]$.

To form our desired topology, we must form a basis collection $\mathcal{B}$. This collection $\mathcal{B}$ contains all open intervals $(a,b)$ in set $X$, if it exists, the smallest element $a_0$ of the half-open interval $[a_0,b)$, and, if it exists, the largest element $b_0$ of the half-open interval $(a,b_0]$. The subbasis for this topology is the finite intersection of open rays:

$\mathcal{B}=\bigcap_{n=1}^i$ (open rays)

Let us consider the Order Topology on $\mathbb{R}\times\mathbb{R}$. Then we get the basis for the topology to be the collection of all open intervals of the for $(a\times b, c\times d)$. We can also think of the product or the real numbers to form a Product Topology.

The Product Topology is defined on the cartesian product $X\times Y$ where $X$ and $Y$ are both topological spaces. To form our basis for this topology, we consider the collection of all products $U\times V$ such that $U$ and $V$ are open subsets of $X$ and $Y$ respectively. So, we can think of each topological space $X$ and $Y$ having their own bases themselves. Therefore, given the basis $\mathcal{C}$ for $X$ and the basis $\mathcal{D}$ for $Y$, we get the following basis $\mathcal{B}$ for the Product Topology $X\times Y$:

$\mathcal{B}=\{C\times D:C\in\mathcal{C} \ \text{and} \ D\in\mathcal{D}\}$.

To consider the $\mathbb{R}\times\mathbb{R}$-example again, we essentially have the product of two open intervals $(a,b)\times(c,d)$ which serves as the basis for the topology. Graphically, it will show the intersections between these open intervals forming the topology on $X\times Y$. However, to form a subbasis for this topology, we must consider functions called projections.

The projection of $X\times Y$ are the functions $\pi_1:X\times Y\to X$ and $\pi_2:X\times Y\to Y$. These functions can be defined by equations such that $\pi_1(x,y)=x$ and $\pi_2(x,y)=y$. If we take $U$ to be an open subset of $X$, then we get the set $\pi_1^{-1}(U)=U\times Y$. The same is with $V$ being an open subset of $Y$: $\pi_2^{-1}(V)=X\times V$. As this might be already obvious, both of these sets are open subsets of $X\times Y$ and their intersection is $U\times V$:

$U\times V=\pi_1^{-1}(U)\cap\pi_2^{-1}(V)$.

Therefore, we define the subbasis collection $\mathcal{S}$ for $X\times Y$ to be

$\mathcal{S}=\{\pi_1^{-1}(U):U \ \text{open in} \ X\}\cup\{\pi_2^{-1}(V) \ \text{open in} \ Y\}$.

We now move to the final “standard” topology: the Subspace Topology.

The Subspace Topology is essentially a subspace $Y$ of a topology $X$ such that

$\mathcal{T}_Y=\{Y\cap U:U\in\mathcal{T}\}$

which defines the topology on $Y$.

We can now define the basis $\mathcal{B}$ for $Y$ such that

$\mathcal{B}_Y=\{B\cap Y:B\in\mathcal{B}\}$.

From here, it should be quite easy to form our subbasis for this Subspace Topology $Y$ of $X$. So I will leave it as an exercise for the reader.