The discussion revolves around the properties of open and closed sets in metric spaces, particularly focusing on the set A in a metric space (A,d) and the definitions and axioms related to topologies. Participants explore the implications of these definitions and the nature of sets within metric spaces.
Participants express differing views on the nature of open and closed sets, particularly regarding the definitions and axioms of topologies. There is no consensus on whether the set of closed subsets can equally define a topology, and confusion remains about the implications of being both open and closed in metric spaces.
Some participants highlight limitations in understanding the definitions and properties of open and closed sets, particularly when considering subspaces and the implications of boundary points. There is also a noted ambiguity in the axioms of topology as they relate to open and closed sets.
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.
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?
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.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.