Continuous image of a connected set is connected
Continuous maps preserve connectedness.
Continuous image of a connected set is connected: Let be a continuous map between topological spaces. If is connected, then is connected.
A continuous map between topological spaces and is a function such that for every open set , the preimage is open in .
A connected set is a subset of a topological space such that there do not exist disjoint nonempty sets that are open in the subspace topology on and satisfy .