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.




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