🛈 This materials is useful for Unit 1 in CSIR-NET.
Definition: , where
Prove that is a metric on , as an exercise.
We denote it as .
In terms of topological spaces, , where is the power set of
Some Interesting Properties:
- Every function is continuous in .
- Every set is open subset of .
- Every set is closed in .
- is not complete.
Examples of interest
- greatest integer smaller than or equal to x is not continuous in with usual metric, but is continuous in .
- is not open in , but it is open in . In fact, every set is open in
- is closed in , and in .
- Can you find an example of compact set in ?
- Can you characterize convergent sequences in ?
Leave your thoughts in the comments below!