Topological tripod
= Topological tripod
Three intervals joined at one endpoint form a <compact> <connected> metric space. It is <second countable>, <Hausdorff> and <paracompact>, but is not a <topological manifold>. At regular arm points a putative manifold has <dimension> one; deleting the junction from a small neighborhood instead gives three components, incompatible with an interval chart. This local obstruction precedes any question about a smooth structure.