Helping provide intuition and clarity as to what exactly an "open set" is.

Solution 1:

In the euclidian topology, the intuition (and actually very much the definition) for an open set is that they are precisely the sets for which you take a point, and you can sufficiently zoom in such a way that you will find a disk that is centered at the point and contained in the set(this is the ugly description in terms of ε that what you cited says).This means that you can get closer and closer and closer to the boundary, as much as you like, but you will still be able to zoom in enough so that you find a circle that keeps you inside it. One consequence is that in the euclidian setting the open sets have no points on their boundary, so you can draw them in a very complicated way, but they still must retain this property, that when you zoom closely enough, you find the picture I described before. If you did have a point in the boundary, try to think of why you can't do the zooming in thing (hint, however close you get, any disk will still go outside the set)

The great property sets of this kind have, is that when you take an arbitrary union of them, they really still keep the same local picture, so you zoom in, and you find the ball contained in the union. Another good property they have is that if you take a finite intersection of them, it still keeps the same local picture.

At this point, we can actually prove that many other good properties can be described in terms of exactly these 2 about the union and intersection, along with the fact that both the empty set and the whole space satisfy the property of the "zooming in". So we define open sets as a collection of sets that satisfy these 3 properties so that they can do, for the most part, the same things the euclidian topology does, even if they are fundamentally different sets.

You then define the closed sets as the complements of these sets. So you can think of them as exactly the sets that, instead, need to contain their boundary(in the euclidian\metric setting). These sets have properties exactly opposite of the open sets, which is that arbitrary intersections are still closed and finite unions are closed (you can prove these by De Morgan's laws). You can define a topology in terms of closed sets, keeping in mind that you have to choose a collection that satisfies these two opposite properties, and then the open sets will be the complements of closed sets.

I wouldn't think too much about the etimology as you are doing, because open and closed sets sometimes do things that don't really justify their name, it is what it is but it is not important, you should think in terms of the properties they have.

Edit: To answer one of your specific questions. Yes, you do just define who the open sets are, in the euclidian topology you define open sets as the ones that satisfy the "zooming in" property. There are many other topologies, each of which will have different open sets, as long as they have the 3 properties, the couple $(X,Τ)$ where $X$ is the base set, like $\mathbb{R^n}$ , and $T$ is the collection of open sets, will be worthy of being called a Topological space. Every time you change the collection of open sets, you get a different topological space, for example there are an infinity of possible topologies on $\mathbb{R^n}$, many of which are useless, but you can still define them as Topological spaces.

Solution 2:

To do topology, it is often helpful to be familiar with the lingo in the basic theory of metric spaces. This is ultimately for practical reasons: while it is possible to do topology from a purely algebraic viewpoint, there is underlying geometry that is encoded by the choice of terminology, which is borrowed from metric spaces.

Strictly speaking, a topology $T$ on $X$ is a collection of subsets of $X$ so that:

• If $\{T_i\}_{i \in I} \subseteq T$, then $\bigcup_{i \in I} T_i \in T$ for any index set $I$,

• If $\{T_i\}_{i=1}^n \subseteq T$, then $\bigcap_{i=1}^n T_i \in T$ for any $n \in \mathbb{N}$,

• And $\varnothing$ and $X$ are in $T$.

Here's the sticking point: we define the open sets of $T$ to be the elements of $T$.

At first glance, this has nothing to do with metrically open (with respect to a metric $d$) subsets of $\mathbb{R}^n$: a subset $S$ of $\mathbb{R}^n$ is metrically open if and only if for each $s \in S$, there is some $\epsilon > 0$ so that the set:

$$B(x,\epsilon) = \{x \in \mathbb{R}^n : d(x,s) < r\}$$

i.e., the open ball centered at $s$ of radius $\epsilon$, is fully contained in $S$. Heuristically, every point of $S$ has "a little room to wiggle around in" while still staying in $S$. That is why the red disc in your image is open. Notably, the open ball is metrically open.

In fact, it is a theorem of metric geometry that every open set in a metric space can be expressed as a union of open balls. Since a union of open balls is metrically open, this is a characterization of the metrically open subsets of a metric space.

Thus the transition between metrically and topologically open goes one way: every metric induces a topology. The set:

$$T_d = \left\{\bigcup_{i\in I} B(x_i,\epsilon_i) : x_i \in \mathbb{R}^n, \epsilon_i > 0, I \text{ an index set}\right\}$$

i.e., the collection of all unions of open balls, satisfies the axioms of topology on $\mathbb{R}^n$! If $d$ is the Euclidean metric, then $T_d$ is the Euclidean topology. We call this the metric topology induced by $d$. Thus we get the direct reason why open sets were named open sets: metrically open sets are topologically open in the metric topology.

Since topology was first done on metric spaces, this gives the historical reason why the sets in the topology are called open; in the metric topology, they are metrically open!

There are many topological terms understood in terms of metric spaces. Among them are vital concepts such as connectedness and compactness. Abstractly, neither of these actually look like what the term sounds like, but again that is because the origin of the term is rooted in metric geometry, and a connected set in the metric topology is indeed "in one piece", and a compact set in the metric topology is indeed metrically compact.

We could certainly rename the sets once we get far away from metric spaces. However, thinking about the sets as "open" is a valuable intuition pump: although most of the geometric properties are lost in the case of a abstract topological space, "openness" captures a sense of "nearness" which is retained.