This is a follow up of this question. Let $M$ be a smooth manifold and let $C \subseteq M$ be a closed, simply connected and locally path connected subset. Is it possible to find an open, simply connected, neigbourhood of $C$?
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freakish
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If $C$ is compact (or just has compact components) then the answer is yes; this is because for locally connected spaces the component decomposition of $C$ is discrete, so it's sufficient to show it for each component. Thus we're dealing with Peano continua, and their shape is completely determined by their Cech cohomology (especially, by the nerve system induced from the basis for $M$). If the components are non-compact then I'm not sure. – John Samples Apr 06 '21 at 19:58