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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.
This is the main functorial property behind connectedness, and it combines naturally with facts about connected sets in (see connected subsets of R are intervals) to identify images of connected domains as intervals.
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 .
Connected subsets of are intervals: View with its usual topology (for example, the one induced by the standard metric). If is connected, then is an interval: whenever with and satisfies , one has . Conversely, every interval in is connected.
This classification is frequently combined with continuous images preserve connectedness to identify images of connected sets under real-valued continuous functions.