Separation and connectedness

Concepts

• Hausdorff space 10 exercisesA topological space X is Hausdorff if for every pair of distinct points x and y there exist disjoint open sets Uni x and Vni y.
• Connected space 10 exercisesA topological space X is connected if it cannot be written as the union of two disjoint nonempty open sets U and V.
• Continuous image of a connected space 10 exercisesIf f:X→ Y is a continuous function between topological spaces and X is connected, then the image f(X) is a connected subspace of…
• Path-connected space 10 exercisesA space X is path-connected if every pair of points a,b∈ X is joined by a path: a continuous function γ:[0,1]→ X with γ(0)=a and…
• Connected component 10 exercisesThe connected component C(x) of a point x of a space X is the largest connected subset A⊆ X containing x.