Open sets are defined

as those sets which contain an open ball around each of their points.

Since I started learning topology, I have not understood this definition. The reason is the following.

If I take an open set, for example the set defined by $U:=\{x|0<x<1, x\in\mathbb{R}\}=(0,1)$ on $\mathbb{R}$, and then choose a point that is rather near to the boundary of its closure $1$, e.g. $0.9$, then by definition, I can draw an epsilon-ball around this point which is still fully contained in the open set. I choose to take the ball with radius infinitesimally bigger than $0.1*0.9=0.09$ such that the ball extends over $0.99$. Now all the points in that ball must again fulfill the definition that I can draw a ball around them which is contained in the set $U$. So let's take a point of the ball we just drew which is nearer to $1$ than the point we chose first, choose $0.99$.

We can draw the epsilon-ball with radius infinitesimally bigger than $0.1^2*0.9$ around this point such that the ball extends over $0.999$. Now we can repeat this process an arbitrary number of times without ever violating the definition of the open set. But if we really do it an infinite amount of times we eventually reach the point $0.9*(1+0.1+0.01+...)=0.\bar 9=1$ which is not in the open set. I know that someone could counter-argue that as we increase the number of repetitions to an infinite amount, we must choose epsilon balls that have a smaller radius to not violate the definition of the open set. But then again: What would be the biggest radius that we are still allowed to choose?

So I just can't imagine how an open set can "work out", to say it colloquially.
More precisely, the only way how I can imagine how every point can have a surrounding that is again made of points which all again must have a surrounding of points, is to have an infinitely extended Set such as $\mathbb{R}$ itself. In that case I can understand that there is just no limit point (or the limit point is infinity) and therefore the above procedure will not lead to a difficulty in my imagination.
But for every Set with a finite extend (even though it may has uncountably many points and even though there is an isomorphism from $\mathbb{R}$ to $(0,1)$), it just does not fit into my intuition. (I know intuition is not everything but without intuition, topics for me loose a certain sense of beauty.)

Or, from another perspective: Imagine you would drive with a car on the real line.
The car shall have a constant velocity, such that its position is given by the function $x(t):= 2-t$, such that at $t=1$, we reach the boundary of the closure of $(0,1)$, then for $t=1+\epsilon$ with $\epsilon>0$, no matter how small $\epsilon$ is, our car will have entered the open set.
Now my intuitive problem with this is that we somehow entered the open set, without passing its first point because there is NO first point in an open set. Or, put differently, us driving the car (or varying $t$ of $x(t)$) is actually a continuous process because $x(t)$ is continuous. And as we pass all the points of the real line in a continuous manner, we could stop anywhere we want (because $t\in\mathbb{R}$). Now when undergoing the transition of the outside of the open set to the inside of the open set, then, because the open set is ultimately just a collection of uncountably many points, we must somehow reach the first point (in my imagination) of that Set but this is impossible for if there was a first point there would not be a surrounding of points around it that is also contained in the set thus contradicting the definition of the open set. So again I cannot imagine this properly.

If there would be a first point of the open set, then it would have to be an uncountable number of points away from the boundary of the closure (e.g. $1$) but if there are uncountably many points between the boundary of the closure and the first point, then it can't be the first point.
If there is no first point of the open set, I can not imagine how the open set can have a finite extend.

So, put into a different question: What would you see if you would look out of the window of a car that drives on the real line into an open set?
Maybe my unability to understand is also connected to a difficulty of imagining the concept of infinity.

Creative and honest thoughts are very much appreciated!

PS: This is my first question on stackexchange. Please excuse the probably outrageous number of things that could have been done better.

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    $\begingroup$ " What would be the biggest radius that we are still allowed to choose?" I think this is crux of your difficulties. There is none. $\endgroup$ – Callus Nov 4 '16 at 16:25
  • $\begingroup$ there can be a biggest radius if you're putting open balls (as in that quote) instead of closed balls. $\endgroup$ – mercio Nov 4 '16 at 16:27
  • $\begingroup$ @mercio Perhaps I misunderstood the question. In the context of the paragraph, I thought the question intended was "what's the biggest radius that we are allowed to choose that works for all points". $\endgroup$ – Callus Nov 4 '16 at 16:28
  • $\begingroup$ "there can be a biggest radius if you're putting open balls (as in that quote) instead of closed balls." No there isn't. 1 is too big as it will include 1. anything less than 1 is too small. In essense the largest radius is equvilent to "the largest number that is less than 1". There is no such thing. $\endgroup$ – fleablood Nov 4 '16 at 16:40
  • 1
    $\begingroup$ Blink and you'll miss it. But you can not avoid blinking. What is an "instant in time". What is an exact point in space. It's still xeno's arrow paradox. Bottom line: there is not first point in an open set, just as there is no first number that is larger than 0. It may seem weird at first that the is not first number larger than 0, but believe me, it'd be a lot weirder if there were. But that is the gyst of it. If there is a last point in one set, there is no first point not in the set, and vice versa. $\endgroup$ – fleablood Nov 4 '16 at 16:51

The alternative perspective you describe sounds like you are describing Zeno's dichotomy paradox — it's basically the same thing as Zeno's more familiar paradox, but oriented in the opposite direction. The basic argument goes as follows:

In order for Achilles to finish the race, he must first arrive at the halfway point. For Achilles to arrive at the halfway point, he must first cross the quarter way point, and so forth.

Thus, Achilles cannot even begin the race, because that would require completing infinitely many tasks.

Like the more familiar race against the tortoise, this is a (pseudo)paradox of infinite divisibility. The distinguishing property of a nonempty, finite sequence of events is:

  • There is a first event.
  • There is a last event.
  • Every event except the last has an immediate successor.
  • Every event except the first has an immediate predecessor.

The paradox is basically expecting all four of these properties to hold for any sequence of events, despite the fact any infinite sequence will violate one or more of them.

In the dichotomy paradox, it is the first of the four properties that is violated.

To resolve this paradox, you have to let go of your expectation that an infinite sequence of events has all of the same properties of a finite sequence.

And if you wish to learn to analyze problems with infinite sequences such as this, you need to develop a new intuition about how those sorts of things work.

  • $\begingroup$ Thanks for the good answer. I think I will accept it as the answer to the question because your comparison to Zenos paradox is very accurate and you name the characteristics of a finite sequence and how to distinguish it from an infinite sequence which is exactly what I am having difficulties with. Still, the very acceptance of the fact that "There is no first element" or "No smallest element bigger than zero" even though an intervall only consists of points is hard for me. I will think about it. If you have additional literature, please let me know. Thanks again. $\endgroup$ – exchange Nov 5 '16 at 10:03

What would you see if you would look out of the window of a car that drives on the real line into an open set?

Since you seem to find the concept of closed sets more intuitive, it might help to think of open sets as complements of closed sets. Here is an example of driving out of an open set.

Suppose you drove your car on the real line coming from $+\infty$ and there is a lowered crossing gate at point ${0}$. You notice it too late and hit the brakes hard. If the car manages to stop at $\epsilon \gt 0$ before hitting the gate, then you are still in the open interval $(0, \infty)$ and, once you step out of the car, you can verify that there is some room left $(0,\epsilon)$ between the car and the gate. No matter how small that $\epsilon$, you may always think that you had room to spare, and could have hit the brakes a bit less hard so as to stop at, say, $\epsilon/2$, and still not hit the gate. On the other hand, if the car does hit the gate that means it just left the open interval $(0,\infty)$ and reached point $\{0\}$. In this case however there is no last point before the car hit the gate since, if such a point existed, it would mean that there were a "gap" between $\{0\}$ and $(0, \infty)$ but intuition tells us that the real line has no "holes".

  • 1
    $\begingroup$ Thank you very much dxiv for the good answer! I have now accepted the answer of Hurkyl for the post because he mentions the paradox as additional literature to read about and he gives the characteristics I need to think about to solve my difficulties. So his post is maybe useful not only for me. However, personally your post was equally important for me and I was actually surprised that this perspective change could change my intuition. The idea of "no holes" illuminates a little why there can't be a first/last point. Also I liked that you used the idea of the car. So thank you very much! $\endgroup$ – exchange Nov 5 '16 at 10:08

Your procedure doesn't reach a contradiction at all.

You show that if a bounded interval is open, then you can find points in the interval converging to a point outside of the interval.

This is true.

  • $\begingroup$ Yes, I know that. But I wrote "a contradiction in my imagination" because I can just not imagine the transition from a finite to an infinite series of balls, there is somehow a jump from a finite number of decimals like 0.99999...9 to an infinite number of decimals like 0.99999..., which seems not smooth to me but at the same time I can imagine going along the real line in a smooth manner. That is why I rephrased the question using the car. fleablood above addressed this already very well. I am sorry that i can not pose the question in a more precise way. But thanks for the answer! $\endgroup$ – exchange Nov 4 '16 at 17:01
  • $\begingroup$ Elsewhere you say "difficulty imagining" and "cannot imagine properly" and things like that. Being difficult to imagine is not the same as being contradictory. So I think this is too subjective a question. Everyone finds new concepts in math difficult to imagine at first. $\endgroup$ – Thompson Nov 4 '16 at 17:09
  • $\begingroup$ Okay, I edited it and changed the line to "difficulty in my imagination". Thanks. $\endgroup$ – exchange Nov 4 '16 at 17:12

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