Connectedness is continuous image-closed

This article gives the statement, and possibly proof, of a topological space property (i.e., connected space) satisfying a topological space metaproperty (i.e., continuous image-closed property of topological spaces)
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Statement

Suppose $X$ is a connected space and $f:X \to Y$ is a continuous map. Then, $f(X)$, endowed with the subspace topology from $Y$, is a connected space.

Related facts

Weaker facts

• Connectedness is coarsening-preserved

Other related facts

• Path-connectedness is continuous image-closed

Applications

• Intermediate-value theorem: This states that any continuous real-valued function on a connected space that takes the real values $a$ and $b$ with $a < b$, takes all real values in the interval $[a,b]$.

Proof

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