Irreducible
Posted: Sun Jun 19, 2016 12:00 pm
A topological space is said to be irreducible if it is not the union of two proper closed subsets.
For the topological space \(\displaystyle{\left(X,\mathbb{T}\right)}\), the following are equivalent :
1. The topological space \(\displaystyle{\left(X,\mathbb{T}\right)}\) is irreducible.
2. Every pair of nonempty open subsets has nonempty intersection.
3. Every nonempty open subset is dense.
For the topological space \(\displaystyle{\left(X,\mathbb{T}\right)}\), the following are equivalent :
1. The topological space \(\displaystyle{\left(X,\mathbb{T}\right)}\) is irreducible.
2. Every pair of nonempty open subsets has nonempty intersection.
3. Every nonempty open subset is dense.