open sets in finite metric space and discrete metric space

Question

How to find all the open sets in a discrete metric space?
And I have one more question
What are all the the open sets in a finite metric space?

https://math.stackexchange.com/questions/132937/show-that-for-a-finite-metric-space-a-every-subset-is-open — user202723, Apr 24 '17 at 07:44

score 1 · Answer 1 · answered Apr 24 '17 at 07:56

1

you know the definition of discrete metric ? $d(x,y)=1, x\neq y$ and $d(x, y)=0, x=y$, using this definition, Now $S(x, r)=\{y : d(x, y)<r\}$. You will found your answer.

answered Apr 24 '17 at 07:56

MSMM

301

i.e. every singletone set is open – MSMM Apr 24 '17 at 07:58

score 0 · Answer 2 · answered Apr 24 '17 at 09:03

Show that every singleton of a metric space with the discrete metric is open. Now as every subset is a union of singletons and a union of open sets is open, conclude that a subset is ...
Show that a finite metric space is discrete, that is every singleton is open. Conclude what the open sets are.

Does every finite metric space have the discrete metric?

open sets in finite metric space and discrete metric space

2 Answers2