Topology: An Inquiry-Based Approach

Consider the interval $(a,b)$ in $\mathbb{R}$ using the Euclidean metric. If $m=\frac{a+b}{2}\text{,}$ then $(a,b)=B(m,\frac{b-a}{2})\text{,}$ so every open interval is an open ball. As an open ball, an open interval $(a,b)$ is a neighborhood of each of its points. This is the foundation for the definition of an open set in a metric space.

Recall that we defined a subset $N$ of $X$ to be neighborhood of point $a$ in a metric space $(X,d)$ if $N$ contains an open ball $B(a,ϵ)$ for some $ϵ>0\text{.}$ We saw that every open ball is a neighborhood of each of its points, and we will now extend that idea to define an open set in a metric space.

So, by definition, any open ball is an open set. Also by definition, open sets are neighborhoods of each of their points. Open sets are different than non-open sets. For example, $(0,1)$ is an open set in $\mathbb{R}$ using the Euclidean metric, but $[0,1)$ is not. The reason $[0,1)$ is not an open set is that there is no open ball centered at $0$ that is entirely contained in $[0,1)\text{.}$ So $0$ has a different property than the other points in $[0,1)\text{.}$ The set $[0,1)$ is a neighborhood of each of the points in $(0,1)\text{,}$ but is not a neighborhood of $0\text{.}$ We can think of the points in $(0,1)$ as being in the interior of the set $[0,1)\text{.}$ This leads to the next definition.

As we will soon see, open sets can be characterized in terms of interior points.