How can a metrix space be open and closed?

In summary, a metrix space is an open set if and only if its complement is open. A metrix space is closed if and only if its complement is open.
  • #1
soopo
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Homework Statement


How can a metrix space be open and closed?
 
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  • #2
Just getting into this stuff, but i'll have a crack

Do you mean metric space?

Take Rn (in Rn) with normal distance metric, clearly any point in the space has a neighbourhood within Rn impling it is an open set

Definition of closed is that the complement of the set is open.

What is the compmelent of Rn? the empty set, which is both open & closed. As its complement is open Rn is closed

A more geometric example might be the 2 sphere, S2, in R3, clearly any point on the sphere has an open neighbourhood implying S2 is open. Look at the complement of the set, R3 / S2. This is clearly also open (though disjoint) meaning S2 in R3 is both open & closed.
 
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  • #3
S^2 in R^3 is closed. It's not open. A neighborhood of a point on the sphere will include points off the sphere as well. For S^2 to be open, the neighborhood would have to be contained in S^2.
 
  • #4


Dick said:
S^2 in R^3 is closed. It's not open. A neighborhood of a point on the sphere will include points off the sphere as well. For S^2 to be open, the neighborhood would have to be contained in S^2.

Do you mean that empty set is the only one which can be closed and open at the same time?
 
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  • #5
All metric spaces are both open and closed as subsets of themselves. The most general definition of "topology" requires that it include both the entire space and the empty set and other definitions (for metric spaces for example) satisfy that. A set is closed if and only if its complement is open. Since the complement of the entire space is the empty set, and the empty set is open, the entire space is also closed.

Those are the only sets that are both open and closed ("clopen") if and only if the space is connected.

For example, if X= (-[itex]\infty[/itex], -1] U [1, [itex]\infty[/itex]), the real numbers with the open interval (-1, 1) removed, with the "usual" metric |x- y|, then X is not connected and X itself is both open and closed, the empty set is both open and closed, (-[itex]\infty[/itex], -1] is both open and closed, and [1, [itex]\infty[/itex]) is both open and closed.
 
  • #6
cheers - i think i get confused between the set & the set its embedded in, ie the set inducing the topology & so the open sets ...

So if i understand on HallsofIvy comment, it all has to do with which set you choose to induce the topology

So, if we take the toplogy induced by the set X = (-inf, -1] U [1, inf), X is open & closed by definition. This gives us the subsets with the properties as described by Halls of Ivy

But if we take the topology induced by R^1

then
C = (-1,1) is open as every points neighbourhood in R^1 is contained in C

-> X = (-inf, -1] U [1, inf) is closed

But the for the point 1 & -1 in X, the neighbourhood in R^1 is not contained in X, so X is not open

So in R^1 the set X is closed & not open

still a bit light on the fundamentals, so does this sound about right? sounds consistent with what Dick said as well?
 
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  • #7
You may want to think about whether you actually have trouble believing that sets can have the properties of an open set and a closed set at the same time, or whether you've convinced yourself (quite understandably) that something which is open can't be closed. The choice of names is unfortunate here- if they were called "O-sets" and "C-sets" would you still wonder how an O-set could be a C-set?
 
  • #8
by the way sorry for hijacking your forum soopo, but any confirmation on my last post would be good though...
 
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  • #9
It's not so much what the topology of the subset as what is the whole metric space. E.g. S^2 with the topology induced by R^3 IS S^2 with the usual topology. S^2 as a subset of R^3 is not open. If you throw away the R^3/S^2 part, then S^2 is the whole space. So it is open.
 
  • #10
thanks for the reply Dick, i know my terminology is a little sloppy, bit of work to do i think...
 
  • #11
S'ok. Trying to help is good, learning from correction is better. That's how I did it.
 

Related to How can a metrix space be open and closed?

1. What is the definition of an open and closed metric space?

An open metric space is a set where every point has a neighborhood that is completely contained within the set. A closed metric space is a set where every limit point of the set is also contained within the set.

2. How can a metric space be open and closed at the same time?

This is not possible. A metric space can either be open or closed, but not both simultaneously. A set cannot contain all of its limit points and at the same time have every point contained in a neighborhood within the set.

3. Can a metric space be neither open nor closed?

Yes, a metric space can be neither open nor closed. This occurs when a set contains some but not all of its limit points. In this case, the set is called a semi-closed or semi-open set.

4. How can we determine if a metric space is open or closed?

To determine if a metric space is open or closed, we can look at the complement of the set. If the complement is open, then the original set is closed. If the complement is closed, then the original set is open. Alternatively, we can also look at the limit points of the set. If all limit points are contained within the set, then the set is closed. If some limit points are not contained within the set, then the set is open.

5. What are the practical applications of open and closed metric spaces?

Open and closed metric spaces have many applications in mathematics and science, particularly in topology and analysis. They are also used in fields such as physics, engineering, and computer science. For example, in physics, open and closed metric spaces are used to describe the properties of space and time. In computer science, they are used to analyze algorithms and data structures. In engineering, they are used to optimize systems and processes.

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