Topologies on product spaces of $mathbb{R}$ and their relationships

Posted 2021-02-02 peabody

In this post, I will summarise several topologies established on the product spaces of (mathbb{R}), i.e. (mathbb{R}^n), (mathbb{R}^{omega}) and (mathbb{R}^J), as well as their relationships.

Topologies on product spaces of (mathbb{R})

1. Topology induced from the euclidean metric (d) on (mathbb{R}^n), where for all (vect{x}, vect{y} in mathbb{R}^n),
   [
   d(vect{x}, vect{y}) = left( sum_{i=1}^n (x_i - y_i)^2 ight)^{frac{1}{2}}.
   ]
2. Topology induced from the square metric ( ho) on (mathbb{R}^n), where for all (vect{x}, vect{y} in mathbb{R}^n),
   [
   ho(vect{x}, vect{y}) = max_{1 leq i leq n} abs{x_i - y_i}.
   ]

3. Product topology on (mathbb{R}^J): its basis has the form (vect{B} = prod_{alpha in J} U_{alpha}), where each (U_{alpha}) is an open set in (mathbb{R}) and only a finite number of them are not equal to (mathbb{R}).

   Specifically, when (J = mathbb{Z}_+), the product topology on (mathbb{R}^{omega}) can be constructed.

4. Box topology on (mathbb{R}^J): its basis has the form (vect{B} = prod_{alpha in J} U_{alpha}), where each (U_{alpha}) is an open set in (mathbb{R}).

   Specifically, when (J = mathbb{Z}_+), the box topology on (mathbb{R}^{omega}) can be constructed.

5. Uniform topology on (mathbb{R}^J): it is induced by the uniform metric (ar{ ho}) on (mathbb{R}^J), where for all (vect{x}, vect{y} in mathbb{R}^J),
   [
   ar{ ho}(vect{x}, vect{y}) = sup_{alpha in J} { ar{d}(x_{alpha}, y_{alpha}) }
   ]
   with (ar{d}) being the standard bounded metric on (mathbb{R}).

   Specifically, when (J = mathbb{Z}_+), the uniform topology on (mathbb{R}^{omega}) can be obtained.

   When (J = n), the topology induced from the metric (ar{ ho}) on (mathbb{R}^n) is equivalent to the topology induced from the square metric ( ho).

6. Topology induced from the metric (D) on (mathbb{R}^{omega}), where for all (vect{x}, vect{y} in mathbb{R}^{omega}),
   [
   D(vect{x}, vect{y}) = sup_{i in mathbb{Z}_+} left{ frac{ar{d}(x_i, y_i)}{i} ight},
   ]
   which is transformed from the uniform metric (ar{ ho}) by suppressing its high frequency component.

   Specifically, when (J = n), the topology induced from the metric (D) is equivalent to the topology induced from the metric (ar{ ho}) and hence is also equivalent to the topology induced from the square metric ( ho).

N.B. In the definitions of product topology and box topology for (mathbb{R}^J) as above, the openness of (U_{alpha}) in (mathbb{R}) is with respect to the standard topology on (mathbb{R}), which does not require a metric to be induced from but only depends on the order relation on (mathbb{R}).

Relationships between topologies on product spaces of (mathbb{R})

According to Theorem 20.3 and Theorem 20.4, the following points about the relationships between topologies on product spaces of (mathbb{R}) are summarised.

1. On (mathbb{R}^n): Topology induced from ( ho) (Leftrightarrow) Uniform topology induced from (ar{ ho}) (Leftrightarrow) Topology induced from (D) (Leftrightarrow) Product topology (Leftrightarrow) Box topology.
2. On (mathbb{R}^{omega}): Topology induced from (D) (Leftrightarrow) Product topology (subsetneq) Uniform topology induced from (ar{ ho}) (subsetneq) Box topology.
3. On (mathbb{R}^J): Product topology (subsetneq) Uniform topology induced from (ar{ ho}) (subsetneq) Box topology.

It can be seen that the finite dimensional Euclidean space (mathbb{R}^n) has the most elegant property, where all topologies are equivalent.