Topology: An Inquiry-Based Approach

Recall that we defined a limit point of a set $A$ in a metric space $X$ to be a point $x\in X$ such that every neighborhood of $x$ contains a point in $A$ different from $x\text{.}$ Since we have defined neighborhoods in topological spaces, we can make the same definition.

The set $A^{\prime }$ of limit points of $A$ is called the derived set of $A\text{.}$

In metric spaces, a set is closed if and only if it contains all of its limit points. So the corresponding result in topological spaces should be no surprise.

In metric spaces we saw that limit point of a set is the limit of a sequence of points in the set. To explore this idea in topological spaces, we define sequences in the same way we did in metric spaces.

We use the same notation and terminology related to sequences as we did in metric spaces: we will write $(x_n)$ to represent a sequence $f\text{,}$ where $x_n=f(n)$ for each $n\in {\mathbb{Z}}^{+}\text{.}$ We can’t define convergence in a topological space using a metric, but we can use open sets. Recall that a sequence $(x_n)$ in a metric space $(X,d)$ converges to a point $x$ in the space if, given $ϵ>0$ there exists a positive integer $N$ such that $d(x_n,x)<ϵ$ for all $n\ge N\text{.}$ In other words, every open ball centered at $x$ contains all of the entries of the sequence past a certain point. We can replace open balls with open sets and make a similar definition of convergence in topological spaces.

If a sequence $(x_n)$ converges to a point $x\text{,}$ we call $x$ a limit of the sequence $(x_n)\text{.}$

The result of Activity 13.4 is that sequences do not behave in topological spaces as we would expect them to. Consequently, sequences do not play the same important role in topological spaces as they do in metric spaces. However, the concept of limit point is important, as are the notions of boundary and closure in topological spaces.