The K-Topology

Recall from the Topological Spaces page that a set $X$ an a collection $\tau$ of subsets of $X$ together denoted $(X, \tau)$ is called a topological space if:

• $\emptyset \in \tau$ and $X \in \tau$, i.e., the empty set and the whole set are contained in $\tau$.

• If $U_i \in \tau$ for all $i \in I$ where $I$ is some index set then $\displaystyle{\bigcup_{i \in I} U_i \in \tau}$, i.e., for any arbitrary collection of subsets from $\tau$, their union is contained in $\tau$.

• If $U_1, U_2, ..., U_n \in \tau$ then $\displaystyle{\bigcap_{i=1}^{n} U_i \in \tau}$, i.e., for any finite collection of subsets from $\tau$, their intersection is contained in $\tau$.

We will now look at an important topology known as the $K$-topology.

Definition: Consider the set of real numbers $\mathbb{R}$ and let $K = \left \{ \frac{1}{n} : n \in \mathbb{N} \right \}$. The $K$-Topology of $\mathbb{R}$ is the collection of subsets $\tau$ such that: 1) $\emptyset \in \tau$. 2) All intervals of the form $(a, b)$ where $a < b$ are in $\tau$. 3) All sets of the form $(a, b) \setminus K$ are in $\tau$. 4) All unions of the sets from (1) - (3).

It's rather difficult to verify the $K$-topology at this point and time, so we will only state it to be a topology and verify it later.