General Topology – Continuous Image of a Locally Connected Space That Is Not Locally Connected

connectednessexamples-counterexamplesgeneral-topology

The question is pretty much in the title, I'm looking for an example of a locally connected space and continuous mapping such that the image is not locally connected.

Thanks!

EDIT: Corrected the phrasing to the intended meaning.

Best Answer

Boring example: any space $X$ is the continuous image of the discrete topology on $X$ (using the identity and noting that any function with a discrete domain is continuous). A discrete space is trivially locally connected (all singleton sets). Now let $X$ be any non-locally connected space.

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