Homework Help: Defining Topological Spaces help

1. Feb 28, 2012

ikenmike05

1. The problem statement, all variables and given/known data

Let ℝ be set of real numbers. Which of the following collection of subsets of ℝ defines a topology in ℝ.

a) The empty set and all sets which contain closed interval [0,1] as a subset.

b)R and all subsets of closed interval [0,1].

c)The empty set, ℝ and all sets such that A not subset of [0,1] and [0,1] not subset of A.

Determine if obtained topology is connected and Hausdorff.

3. The attempt at a solution

Im not sure how to interpret subsets of the closed interval [0,1] and this doesnt seem like it would be an open set.

2. Feb 28, 2012

HallsofIvy

"subset of [0, 1]" means exactly what it says- things like [1/2, 3/4], (1/3, 4/5), etc. This question does not ask it they are "open sets" in terms of the "usual topology", it asks whether the collection of all sets forms a topology.

You should recall that to be a "topology" a collection of subsets of set X must
1) contain X itself.
2) contain the empty set
3) contain the union of any sub-collection of these sets
4) contain the intersection of any finite sub-colection of these sets