Why Use Open and Closed Sets in Mathematical Proofs?

[wayneckm]

Hello,

I am a newbie of this forum. Nice to meet you all.

When I read mathematical proofs or whatever, I always come across open/closed set but I simply have no clue on why and when should one start with them. For example, "Assume the parameter \theta takes values in \Theta, where \Theta is an open set, ...", so what's the motivation for such assumption? It is just arbitrarily up to someone's preference? or somehow this kind of things may fascilitate proofs/key steps afterwards?

I always try to understand people's work on what's their motivation on doing/defining/assuming something.

Thanks.


Wayne

 

[dx]

Open sets are meant to represent the notion of a set in which for every point x, all points "sufficiently close to x" are also contained in the open set.

So for example, if in some topological space X, the set {x} is open, then it means that the point x is 'isolated', i.e. there are no other points topologically close to x.

 

[RedX]

Open sets are meant to represent the notion of a set in which for every point x, all points "sufficiently close to x" are also contained in the open set.

So for example, if in some topological space X, the set {x} is open, then it means that the point x is 'isolated', i.e. there are no other points topologically close to x.

How do we define open sets in a topological space? In a metric space it is simple as you can define an open ball when given a metric (distance function).

I read somewhere that for a manifold you determine if a set is open by defining a one-to-one map from that set to Euclidean space, and if the image in Euclidean space is open, then the original set is open. Is this correct?

Assuming this is correct, say your manifold is a 1-dimensional string. For example, put the endpoints of the string at x=0 and x=L in the xy-plane, and allow the curve to wiggle in the y-direction. Is it even possible to find a one-to-one map of the string onto R¹ such that the image is open in R¹? To me this can't be done because of the endpoints: it would be like mapping a closed set onto an open set which I don't think is possible.

 

[dx]

How do we define open sets in a topological space? In a metric space it is simple as you can define an open ball when given a metric (distance function).

In a metric space, the fundamental structure is the metric function, and the topology is 'induced' by the metric. On the other hand, topological spaces don't have any other structure than their topology, so the topology is simply defined by designating certian subsets as 'open', subject to some conditions.

I read somewhere that for a manifold you determine if a set is open by defining a one-to-one map from that set to Euclidean space, and if the image in Euclidean space is open, then the original set is open. Is this correct?

Assuming this is correct, say your manifold is a 1-dimensional string. For example, put the endpoints of the string at x=0 and x=L in the xy-plane, and allow the curve to wiggle in the y-direction. Is it even possible to find a one-to-one map of the string onto R¹ such that the image is open in R¹? To me this can't be done because of the endpoints: it would be like mapping a closed set onto an open set which I don't think is possible.

Look at the section 'manifolds with boundary' in this page: http://en.wikipedia.org/wiki/Topological_manifold

 

[zhentil]

How do we define open sets in a topological space? In a metric space it is simple as you can define an open ball when given a metric (distance function).

This puts the cart before the horse. You don't have a topological space until you've defined the open sets.

 

[Werg22]

Sounds like you need to read on point-set topology and elementary general topology.

 

[mordechai9]

This is a question I kind of have myself. I'd like to see some more detailed responses from others... Sorry I'm a math newb.

In geometry of curves, they often talk about curves being defined on an open set. In other words, if we think of a plane curve, we would have that $$\alpha : I \rightarrow \mathbb{R}^2$$ then $$I$$ should be an open interval, e.g., $$I=(a,b) \subset \mathbb{R}$$ with $$a < b$$. I think the reason they do this is because the derivative of the curve does not exist at the endpoints of a closed interval, where the curve must come to a stop. Since curves must have finite speed to make sense, then they require the interval of definition to be open. For an open interval, you can always choose a point in the interval close to the endpoints where the curve still has finite speed (nonzero derivative).

There are other examples though, like you alluded to in your question. However, I'm having trouble thinking of other examples, so maybe it would be helpful for you to ask another, more specific question, or for someone with a better understanding to chip in here.

 

[element4]

(...)

For example, "Assume the parameter \theta takes values in \Theta, where \Theta is an open set, ...", so what's the motivation for such assumption? It is just arbitrarily up to someone's preference? or somehow this kind of things may fascilitate proofs/key steps afterwards?

(...)

It is of course not an arbitrary assumption. If you start with a set containing some objects, you can define the notion "open set" and if your definition satisfies certain http://en.wikipedia.org/wiki/Topological_space#Definition", your set is called a topological space. As a very simple example take $$\mathbb{R}^2$$, then you can use the metric $$\|x-y\|$$, $$x,y\in\mathbb{R}^2$$, to define open sets (and this makes $$\mathbb{R}^2$$ a topological space). In general you can define much more general and abstract topologies on different sets (and its not always unique).

What is this good for? Well, you can use your topology to define notions like http://en.wikipedia.org/wiki/Contin...tinuous_functions_between_topological_spaces".)

It's not possible to give a general answer to your question. But when you see an assumption of openness, then the proof probably depends on some sort of topology of some set, continuity of some functions or maybe convergence of some limiting procedure. There are many possibilities. If you have some specific proof, maybe we can try to point out why the assumptions is important in that case...?

I hope this helps a little.

 

[Tac-Tics]

Open and closed sets are based on the idea of a metric space. Most topology textbooks explain them pretty well. You start off with a distance function d(x, y) that defines the distance between two points. You define two new objects: open balls and closed balls.

The open ball centered at x with radius r is the set of all points y where d(x, y) < r.

The closed ball centered at x with radius r is the set of all points y where d(x, y) <= r.

In R^1, a ball is an interval. In R^2, balls look like circles. In R^3, they look like spheres. But you can also define suitable distance functions for functions spaces and other weird spaces.

Then, you can define open and closed sets in terms of balls.

An open set is one that can be written as the (possibly infinite) union of open balls.

A closed set is one where its complement is open.

(Note that closed and open are not opposites. A set can be closed AND open. And another set might be neither closed NOR open).

So when would you use an open set? It's a bit hard to explain in words, but the proofs are usually very simple and elegant, and you could say it "just works" that way.

To me, an open set is a "fuzzy" set. If I have a point x and I ask you to give me an open set containing x (a "neighborhood of x"), no matter how small of an open set you give me, I can wiggle x while keeping it inside of the set. It's that wiggle room that is the basis of continuity.

Something to chew on -- Continuous functions are a natural fit for science. Why? Because the nature of measurement is inexact. Regardless of what our instruments say, the actual value of a measurement is going to be a little off. If our instrument has a needle which records our value on graph paper, for instance, vibrations in the room will cause the needle to vibrate and give us slight errors in our measurements. Put anther way, vibrations in the room might cause our needle to wiggle.

The motivation for open sets is we will always have errors in our measurements. The definition of continuity is exactly that -- if you want to keep your errors under a given threshold (keep the wiggling contained within a small open set), you need to make a machine capable accurate enough to detect it (keep the measurement errors contained within another small open set). That is, a continuous function is one where open sets are mapped to by open sets.