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1. ### Do open sets in R^2 always have continuous boundaries?

An open set can be written as the union of multiple disjoint non-empty open sets if and only if it is disconnected. Therefore another way of phrasing this statement is: Any simply-connected connected open set U can be written as an open set of type 1 or type 2. To see that there exists...
2. ### Natural examples of metrics that are not Translation invariant.

I just looked up "metric spaces" on wikipedia and two examples stood out: 1) Give the positive real line the metric d(x,y) = |log(x/y)| 2) In Euclidean space suppose that instead of considering the direct distance from x to y, we want to travel via 0, then the distance is given by: d(x,y) = |x|...
3. ### Sup of a sequence

I doesn't look like you really understand sup. The n below sup is meant as a variable, and sometimes we have restrictions such as \sup_{n\geq 5} x_n which means we consider sup of the sequence x_5,x_6,x_7,\ldots. Therefore your first two statements do not seem to make sense (at least with...
4. ### Relationship between seminormed, normed, spaces and Kolmogrov top. spaces

For the => direction you are on the right track. You want an open ball B with center x, but with radius small enough that y is not in B. d(x,y) seems to be the only number you have to work with so try choosing a radius based on that. If you can find a radius such that y is not in B, then you are...
5. ### Functions that separate points

(iii) is not the same kind of separation as that in (i) and (ii). To see the difference just take the real line and let the family of seminorms be the set consisting of just the ordinary absolute value norm. Then clearly (i) holds because |x|=0 implies x=0, but (iii) does not hold because 1\not=...
6. ### How is (0,1) not compact?

{0} and {1} are not open sets so you only provided a cover with no finite subcover, but compactness only guarantees that an OPEN cover has a finite subcover. Otherwise no infinite set could be compact because we could just cover it by all it's one-element subsets.