📂Topology

Trivial Topology and Discrete Topology

Definition ¹

When a set $X$ is given, if we endow it with the trivial topology $\left\{ \emptyset , X \right\}$, then that space is the smallest space and is called the trivial space. Conversely, if we endow it with the discrete topology $\mathscr{P}(X)$, then that space is the largest space and is called the discrete space.

Sierpinski Space

If the topology of $S : = \left\{ 0, 1 \right\}$ is $\mathscr{T} : = \left\{ \emptyset , \left\{ 1 \right\} , \left\{ 0, 1 \right\} \right\}$, then $S$ is called the sierpinski space.

The Sierpinski space is a space with only two elements whose topology is neither the trivial topology nor the discrete space. The only difference is whether the chosen element is $0$ or $1$, but since they are essentially the same, there is no point in distinguishing them. When specified with the topology above, $\left\{ 1 \right\}$ is open and $\left\{ 0 \right\}$ is closed.

Derived Set of the Sierpinski Space

$$ \emptyset ' = \emptyset \\ \left\{ 1 \right\} ' = \left\{ 0 \right\} \\ \left\{ 0 \right\} ' = \emptyset \\ \left\{ 1,0 \right\} ' = \left\{ 0 \right\} $$

Perhaps because of the appearance of $\emptyset$, which looks like a combination of $0$ and $1$, this is an example in which mathematical beauty is maximized. If you are encountering topology for the first time, I hope you will try proving directly whether the above results are true, as an exercise. Without any special technique, you can easily verify them by simply computing and listing according to the definition.