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

Question

I have read What does "open set" mean in the concept of a topology? and studied the open set wiki, as well as reading through the open set definition of a topological space, but yet I am still not crystal clear on what exactly an open set actually is (and this concept of "open set" is required to be understood in order to dig into many areas of topology and elsewhere, so I am stuck currently, trying to resolve what open set actually means).

First, as one of the other answerer's in the first link points out:

In an abstract topological space, "open set" has no definition!

Which I think Wikipedia confirms, by saying:

A topological space is a set on which a topology is defined, which consists of a collection of subsets that are said to be open, and satisfy the axioms given below.

So basically, it sounds like a topological space is by definition composed of open sets? I am not sure.

The only example I really 80% get is the image:

The blue circle represents the set of points (x, y) satisfying x2 + y2 = r2. The red disk represents the set of points (x, y) satisfying x2 + y2 < r2. The red set is an open set, the blue set is its boundary set, and the union of the red and blue sets is a closed set.

That kind of makes sense, that the open set is everything except the boundary, but even then not really. Perhaps, is there another term we could pick other than calling it "open" (and "closed"), maybe taking inspiration from German or French or another language?

Anyways, we have this now to start (on the open sets wiki page):

A subset ${\displaystyle U}$ of the Euclidean n-space $R^n$ is open if, for every point x in ${\displaystyle U}$, there exists a positive real number $ε$ (depending on x) such that a point in $R^n$ belongs to ${\displaystyle U}$ as soon as its Euclidean distance from x is smaller than $ε$. Equivalently, a subset ${\displaystyle U}$ of $R^n$ is open if every point in ${\displaystyle U}$ is the center of an open ball contained in ${\displaystyle U}$.

Here I try to draw the first part out:

The green is the $R^n$, the blue is the open set $U$, the x is a random point in $U$, and the yellow is a point in $U$ because $ε$ is within a certain size. That's as much as I can understand from that sentence.

But even then, it doesn't make sense to me what we are talking about. And this is talking about in Euclidean space with the real numbers, so we are not even in the abstract realm of topological spaces yet.

Can you please try to provide an intuition closer to this? (which still doesn't make it easy to grok, but it is getting there)

What exactly is an open set? First if you could do better on my illustration (describing it better is all, no need to make an illustration unless it would help). And then but the main thing is, how to intuit the meaning of the topological notion of open set, given that you are defining the topology as being composed of open sets! Can you not compose a topology of closed sets? Either way, how do you conceptualize of the abstract notion of open sets?

Other resources that didn't quite help:

• https://mathoverflow.net/questions/19152/why-is-a-topology-made-up-of-open-sets
• Motivation for the concept of "open set" in topology

Answer 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.

Answer 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.

Answer 3

To study topology before one has seen metric spaces (or at least Euclidean spaces) is like studying multiplication before addition: it is very likely not to make much sense. When one studies metric spaces an open set $A$ in a space $M$ is clearly defined: it is a set $A$ such that around every $a\in A$ there is some open ball $B_r(a)=\{x\in M | d(x,a)<r\}$ which is contained in $A$, or in other words, every point of $A$ is an interior point of $A$. So for instance $(0,1)\subseteq\mathbb{R}$ is an open set, but $[0,1)\subseteq\mathbb{R}$ is not open, because every open ball (i.e. interval) centered at $0$ is going to contain points from outside of $[0,1)$.

(Then one goes on to define a closed set as one of two equivalent conditions, either that its complement is open, or that every convergent sequence with elements of the set has limit inside that same set.)

After all this one studies all kinds of properties of open sets, including that the intersection of two (and hence a finite number) of open sets is an open set, and that the union of an arbitrary number of open sets is an open set. Together with the fact that the empty set and the whole metric space are open sets (as follows easily from the definitions), this is exactly what motivates the definition of a topological space where an 'open set' is that which satisfies exactly these conditions.

In a metric space this definition gives us exactly the same old open sets, and in other more general contexts where we are not working with a metric space this turns out to be the most useful and interesting generalization. I highly suggest getting a good grip of the fundamentals of metric spaces before moving on to study topology, otherwise not only the definitions but many of the theorems are going to feel unmotivated as well.