# Can we regard Hausdorff space as a manifold?

Can we regard Hausdorff space as a manifold of class ?(p≥1)

And I want to know the relation among the concept Hausdorff space,metric space,vector space,tangent space and manifold.

What's the common ground between -manifold and smooth manifold?

• Do you mean the other way around? A differentiable manifold is a Hausdorff space, but many topological spaces certainly aren't manifolds. – Christopher A. Wong Feb 23 '14 at 21:31

No, the space $X:=\{(x,y)\in\mathbb{R}^2:xy=0\}$ is Hausdorff when endowed with the subspace topology from $\mathbb{R}^2$. Yet, $X$ does not support any type of manifold structure since it is not locally Euclidean near $(0,0)$.