Connected sets in a topological space

In summary, the definition of Y being connected in a topological space (X, tau) is that you can't find two non-empty, open and disjoint sets whose union is Y.
  • #1
logarithmic
107
0
The definition of Y being connected in a topological space (X, tau) is that you can't find two non-empty, open and disjoint sets whose union is Y.

This doesn't quite make much intuitive sense to me.

For example, consider R with the usual topology. Then clearly, Y= [0,1] union [2,3] is not connected. That means you CAN find two non-empty, open and disjoint, sets whose union is Y. But what are they?

I can't seem to think of 2 open sets in R whose union is [0,1] union [2,3].
 
Physics news on Phys.org
  • #2
The set [0,1] is closed in R, but open in the relative topology of Y (with respect to the usual topology on R). The same is true of [2,3].
 
  • #3
The problem is that your definition is wrong.

A topological space is connected if and only if you cannot find two such open sets. That is not the case for a subset of a topological space. (You can also replace "open" with "closed".)

For a subset, A, of a topological space, X, A is connected if and only if it is not the union of two separated sets.

Sets, U and V, are said to be separated if and only if [itex]\overline{U}\cap V= \phi[/itex] and [itex]U\cap \overline{V}= \phi[/itex], where [itex]\overline{U}[/itex] is its closure.

In your example, we can take U= [0, 1] and V= [2, 3].

Of course, we can always think of a subset of a topological space as being a topological space with the relative topology: A subset of A is open "in A" if and only if it is the intersection of A with a set open in X.

If [itex]A= [0, 1]\cup [2, 3][/itex], then, since [itex][0, 1]= A\cup (-1/2, 3/2)[/itex] and [itex][2, 3]= A\cap (3/2, 7/2)[/itex] both [0, 1] and [2, 3] are open in A (or "open relative to A").

Notice that, in A (or "relative to A") [0, 1] is the complement of [2, 3] and [2, 3] is open, so [0, 1] is also closed. Similarly [2, 3] is also both open and closed in A. It is always true that "connectedness components" of a space (connected sets not properly contained in any connected set) are both open and closed.
 
Last edited by a moderator:
  • #4
Thanks for clearing that up. I'm not sure what your definition of separated sets is meant to say though, I think you made a typo there.
 
  • #5
You are right. For some reason I used \overbar(U) instead of \overline in my LaTex to denote "closure" and it didn't work! I have editted my original post.
 
Last edited by a moderator:

FAQ: Connected sets in a topological space

1. What is a connected set in a topological space?

A connected set in a topological space is a subset of the space where every pair of points can be connected by a continuous curve that lies entirely within the set. In simpler terms, a connected set is a set where all points are "close" to each other in a continuous manner.

2. How is a connected set different from a disconnected set?

A disconnected set in a topological space is a subset of the space that can be separated into two or more disjoint open sets. In other words, it is possible to find a "gap" or "hole" between points in a disconnected set, while a connected set has no such gaps or holes.

3. Can a set be both connected and disconnected?

No, a set cannot be both connected and disconnected in a topological space. If a set is connected, it cannot be separated into two or more disjoint open sets, which is the defining characteristic of a disconnected set. Similarly, if a set is disconnected, it cannot have all points "close" to each other in a continuous manner, which is the defining characteristic of a connected set.

4. Are all subsets of a topological space connected?

No, not all subsets of a topological space are connected. Some subsets may be disconnected, while others may be connected. It ultimately depends on the specific subset and its arrangement of points within the space.

5. How are connected sets important in mathematics and science?

Connected sets are important in mathematics and science because they provide a way to study the continuity and connectedness of points in a space. They also have applications in various areas such as topology, analysis, and physics. Understanding connected sets can also help in solving problems related to continuity, path-connectedness, and compactness.

Similar threads

Back
Top