Understanding Open and Closed Sets in Metric Spaces

In summary, in a metric space (A,d), the set A is both open and closed. This is because any open set in the metric space is contained in A and A is regarded as the full space containing all the necessary points. However, if (A,d) is considered as a subspace of a different metric space, A may not necessarily be open or closed in that space.
  • #1
Bobhawke
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I have some topology notes here that claim that on any metric space (A,d), A is an open set

But surely we can just take a closed set and define a metric on it, like [0,1] in R with normal metric?
 
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  • #2
A set can be both open and closed. The entire space and the empty set are both open and closed in any topological (or metric) space. If these are the only open and closed subsets, then the space is said to be http://en.wikipedia.org/wiki/Connected_space" [Broken].
 
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  • #3
I just read that in a metric space (A,d) the set A is both open and closed but I don't understand why

Further, in my notes it says that we want to define a topology on a set to be the set of all open subsets. It then formally defines a topology to be the set of subsets of A that satisfies:
1. A and the empty set are in T (where T is the topology)
2. Any union of elements of T are in T
3. Any intersection of elements of T are in T

But it seems to be the set of all closed subsets would equally well satisfy all those axioms, since A and the empty set are also closed, any union of closed sets is closed and any intersection of closed sets is closed.
 
  • #4
In a metric space, a set is closed if it contains all of its boundary points. A set in open if it contains none of its boundary points. Since the entire set itself has no boundary points, both of those are true: "all"= "none" so it both open and closed.
 
  • #5
Bobhawke said:
I just read that in a metric space (A,d) the set A is both open and closed but I don't understand why

Further, in my notes it says that we want to define a topology on a set to be the set of all open subsets. It then formally defines a topology to be the set of subsets of A that satisfies:
1. A and the empty set are in T (where T is the topology)
2. Any union of elements of T are in T
3. Any intersection of elements of T are in T

It should be
3. Any finite intersection of elements of T are in T.

But it seems to be the set of all closed subsets would equally well satisfy all those axioms, since A and the empty set are also closed, any union of closed sets is closed and any intersection of closed sets is closed.

The closed sets satisfy similar axioms but the role of intersection and union is reversed: finite unions of closed sets are closed and any intersections of closed sets are closed.
 
  • #6
Bobhawke said:
But surely we can just take a closed set and define a metric on it, like [0,1] in R with normal metric?

Here's an open set in your space:

{ x : d(x,1/2) < 1 }

The set of points strictly less than 1 unit distance from 0.5. It's open, surely you agree? It's also all of [0,1], so that set is an open subset in that metric space you just defined! Note that something like 1.1 is not in the interval [0,1] so isn't in the set I just defined.
 
  • #7
In particular [0, .1) and (.9, 1] are both open sets in [0,1]. they are open neighborhoods in [0,1].
 
  • #8
Bobhawke said:
I just read that in a metric space (A,d) the set A is both open and closed but I don't understand why

Further, in my notes it says that we want to define a topology on a set to be the set of all open subsets. It then formally defines a topology to be the set of subsets of A that satisfies:
1. A and the empty set are in T (where T is the topology)
2. Any union of elements of T are in T
3. Any intersection of elements of T are in T

But it seems to be the set of all closed subsets would equally well satisfy all those axioms, since A and the empty set are also closed, any union of closed sets is closed and any intersection of closed sets is closed.
A is open because any open set in the metric space (A,d) is contained in A and this will imply that any open sphere centred on any point of the space (A, d) will surely be in A. On the other hand, A is closed because (A, d) is regarded as the full space w.r.t the metric d, thus all the necessary points are contained in A. for instance, If x is an arbtrary point of A where A is said to contain all the points needed, then x is a limit point of A or an isolated point of A and it is contained in A. By definition of a closed set, it must contain its limit point. (NB. If a set is considered as a metric space then it can be regarded as the fullspace containing all the points needed). But, If (A,d) is regarded as a subspace of a metric space say (X, d') where d has been restricted by d', then A may niether be closed nor open subset of X with respect to the metric d' restricting d, where (A,d)=(A,d') and (X,d) is not in general equal to (X,d'). Thus as a metric Space in its own right, A is both open and closed.
 

1. What is an open set in a metric space?

An open set in a metric space is a set of points that does not contain its boundary. In other words, for any point in the set, there exists a neighborhood around that point that is entirely contained within the set. This definition is based on the concept of distance between points in a metric space.

2. How are open sets related to metric spaces?

Open sets are a fundamental concept in the study of metric spaces. They allow us to define important properties such as continuity, convergence, and connectedness. In fact, the definition of a metric space itself relies on the concept of open sets.

3. What is the difference between an open set and a closed set?

An open set does not include its boundary, while a closed set does. In other words, a point on the boundary of a closed set is considered to be part of the set, while a point on the boundary of an open set is not. Additionally, every point in an open set has a neighborhood that is entirely contained within the set, while this may not be true for a closed set.

4. How are open sets defined in different topologies?

Open sets are defined differently depending on the topology of the space. In a Euclidean space, open sets are defined using the concept of distance, while in a topological space, open sets are defined based on the structure of the space itself. In general, open sets in different topologies may have different properties and behave differently.

5. Why are open sets important in mathematics?

Open sets are important because they allow us to define important concepts such as continuity and convergence in metric spaces. They also play a crucial role in the study of topological spaces and their properties. Additionally, the concept of open sets has many applications in various fields of mathematics, including analysis, topology, and geometry.

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