I was wondering if the following is true: In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.
Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?
In a topological space, what other kinds of condition can make inf and sup of a subset achievable in itself?
Thanks and regards!
More question:
In an "ordered" (not sure what kinds of order is proper here) topological space, are inf and sup of a subset accumulation points of the subset?
More questions again:
In Euclidean space, are inf and sup for a closed subset inside the subset? Are they accumulation points of the subset? What if in metric space? Thanks!