Main theorem of connectedness

Let $𝑋$ and $𝑌$ be topological spaces and $𝑓 : 𝑋 \to 𝑌$ be a continuous function. If $𝑋$ is (path)connected, then so is $𝑓 (𝑌)$ . #m/thm/topology

Proof for plain connectedness

Without loss of generality, consider a surjection $𝑓 : 𝑋 ↠ 𝑌$ . If $𝑌$ is disconnected, it can be partitioned into open $𝑉 1$ and $𝑉 2$ , wherefore $𝑋$ can be partitioned into open $𝑓 - 1 𝑉 1$ and $𝑓 - 1 𝑉 2$ and is thus disconnected. Alternatively, if $𝑌$ is disconnected then there exists non-constant continuous $𝑔 : 𝑌 ↠ {0, 1}$ , wherefore $𝑓 𝑔 : 𝑋 ↠ {0, 1}$ is nonconstant and continuous. Thus, if $𝑋$ is compact so is its continuous image $𝑓 (𝑋)$ .

Proof for path-connectedness

Given any two points $𝑎, 𝑏 \in 𝑋$ there exists a continuous function $𝑐 : [0, 1] \to 𝑋$ such that $𝑓 (0) = 𝑎$ and $𝑓 (1) = 𝑏$ . Clearly, $𝑓 𝑐$ constitutes a continuous function $𝑓 𝑐 : [0, 1] \to 𝑓 (𝑋)$ , and therefore for any two points $𝑓 (𝑎), 𝑓 (𝑏) \in 𝑓 (𝑋)$ there exists a Continuous path $𝑓 𝑐$ such that $𝑓 𝑐 (0) = 𝑓 (𝑎)$ and $𝑓 𝑐 (1) = 𝑓 (𝑏)$ . Thus $𝑓 (𝑋)$ is path connected.

This is a remarkably rare instance of properties being inherited by images, usually properties are inherited by preïmages.

Corollaries

Connectedness and Path connectedness are topological properties.
The quotient of a connected space is connected.
Connected fibres and quotient implies connected space

#state/tidy | #lang/en | #SemBr