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We introduce the notion of an EILC topos: a topos $\mathcal{E}$ such thatevery essential geometric morphism with codomain $\mathcal{E}$ is locallyconnected. We then show that the topos of sheaves on a topological space $X$ isEILC if $X$ is Hausdorff (or more generally, if $X$ is Jacobson). Furtherexamples of Grothendieck toposes that are EILC are Boolean \'etendues andclassifying toposes of compact groups. Next, we introduce the weaker notion ofCILC topos: a topos $\mathcal{E}$ such that any geometric morphism $f :\mathcal{F} \to \mathcal{E}$ is locally connected, as soon as $f^*$ iscartesian closed. We give some examples of topological spaces $X$ and smallcategories $\mathcal{C}$ such that $\mathbf{Sh}(X)$ resp.$\mathbf{PSh}(\mathcal{C})$ are CILC. Finally, we show that any Booleanelementary topos is CILC.

Toposes over which essential implies locally connected