Closure of sets

>>7673024
Because limit points are not the same thing as boundary points.

>>7673024
This is a topological notion, not something you'd see in a set theory book.

Closed is not defined to mean "not open". A set in a given topology T on a set X is closed if it's complement in a given topology is open. The fact that a set can be both open and closed is trivial since the X and the empty set both must be open sets by definition of a topology, and so since they are eachother's complement, they're also both closed.

>>7673024

Topologically speaking, the definition of closed is that its compliment is open. That means any set with an open compliment is closed. So if you have an open set, if its compliment is open then it is also closed. If you have a set that is not open, and its compliment is not open, then it is neither open nor closed. The problem with "open" and "closed" is that they imply mutual exclusivity, when in reality they're two different qualifications.

>>7673024
that's topology mate

>>7673024
Pretty much what everyone else said. Opened and closed are only meaningful terms when you are dealing with a topological space. As for resources to find this stuff, http://mathworld.wolfram.com/OpenSet.html

I'm more than a little rusty, can someone give me a non-trivial example (neither the null set or its complement) of a set that is both closed and open? A set that's neither is trivial (just glue half of an open circle to half of a closed circle).

Open means that every point within the set contains a neighborhood around that point that is within the set. A simplistic intuition would make it seems like the complement of that would naturally be a closed boundary (a curve of limit points), but you're saying this isn't the case (and I remember that it's not). So what's an example?

>>7673024
I suppose you've first encountered a "closed" or "open" set when reading about metric spaces. There, you usually have a big set X that can be seen as a universal set, and your "closed" set C is a subset of X. The definition of being open uses some epsilon-balls built using the defined metric d. Actually, there is a deeper mechanism behind this - a Topology T is a special set of subsets of X. And an open set is by definition just any set in T, a closed set is any subset of X, so that it's complement is in T. There are 3 conditions (axioms), T has to meet. One of them is that X and the empty set are in T. That means for example X is open and closed at the same time. It's simple as that, you just have to check if A or it's complement is in T to know whether it is open or closed. This is a point when notions in mathematics dont relate to their Meanings in real-life anymore. Okay lets say I only have a metric space, not a topological space, how the hell do I get this Topology T? Well there's a thing called metric-induced Topology, that means if you have a metric d you can build your Topology with a simple rule: a set A is in T iff there are epsilon balls fully enclosed in A around every point of A. That's pretty similar to the actual definition of being open with respect to metric, isn't it? Thats how it approximately works. So if you have a topology T on your X, and a set A happens to be in T together with the complement of A, then you have a set, that is closed and open at the same time.

>>7673079

Consider the discrete topology on X, where every subset of X is open. You can quickly verify this satisfies the definition of a topology. Futhermore, the compliment of any subset Y of X is a subset of X, and therefore open, meaning Y is closed. But Y is also itself a subset of X, so Y is open as well.

Basically, in the discrete topology everything is clopen.

This could still be considered a "trivial" example, however its the easiest one I can think of.

>>7673079
i don't think there's a nontrivial set, open with respect to metric-topology (the one you're probably using even you might not know about it) that also happens to be closed. The example of anon above with discrete topology is a pretty good one, but probably not what you expected.

>>7673095
Oh, right, I always assume that Topologies imply more than they necessarily do. Guess I was a pretty poor student of Topology since the entire point is generalizing metric spaces and not just thinking them all as Euclidean or trivial distortions of Euclidean space.

The definition of disconnectedness is the following:

Let [math]X[/math] be a topological space. If there exist such [math]A, B \subset X[/math] that [math]X=A \cup B[/math], [math]A \neq \emptyset \neq B[/math], [math]A \cap B = \emptyset[/math], and [math]A, B[/math] are open in [math]X[/math], then [math]X[/math] is said to be disconnected.

Let then [math]A[/math] be some proper subset of a disconnected space, and let [math]B[/math] be its complement. Then [math]X=A \cup B[/math], [math]A \neq \emptyset \neq B[/math], [math]A \cap B = \emptyset[/math]. Assume that [math]A[/math] is the set in the definition. Then [math]B[/math] is the complement of an open set, so it is closed, but it is also open (by definition).

>>7673079
Well the thing is that if you think of your space as being one piece, then you will have a hard time because it's not possible (it's actually how connectedness is defined).
However if you take your space to be comprised of two disjoint open pieces then they will both be opened and closed.
For example, in the space [math]\mathbb R\setminus\{0\}[/math] with the topology induced by that of R, the interval [math](0,\infty)[/math] is both open and closed.

>>7673024
A set being closed does not imply that it is not open because there are clopen sets. If the set is not clopen, and it is closed, then that implies that it is not open. What is a clopen set? It is a bullshit way to fix the following "paradox" : A closed set is a set such that its complement in the space X is open. An open set is a set such that, for any point in the set, there exists a neighborhood about the point that lies entirely within the set. The space X is the union of (0,1) and (2,3) (which are each from the real line). Take the set (0,1), is it open or closed? Its complement in the space X is (2,3), which is open, so (0,1) should be closed, but, (0,1) is clearly open. Instead of realizing that the definitions of closed sets are stupid, mathematicians invented the clopen set.

ITT
https://www.youtube.com/watch?v=SyD4p8_y8Kw