Defining Closed, Open, and Compact Sets in R^n

[Garcher]

Homework Statement

How to define closed,, open and compact sets?Are they bounded or not?

Homework Equations

For example {x,y:1<x<2}

The Attempt at a Solution

It's is opened as all points are inner

Can you please say the rule for defining the type of the set? Like for example 1<=(x^2+y^2)<=2?

 

[radou]

You are talking about sets in R^2. In general, before considering topological properties, you first have to specify the topology you're considering. In R^n, the topologies induced by the standard euclidean metric are the same as the product topology. So, I'll assume you're given the standard euclidean metric and looking at it's corresponding metric topology.

Your first set is open, since for any point in the set you can find an open ball around that point belonging to the set. Can you do this for the second set?