Topological Manofold is Hausdorf, Second Countable, Locally Homeomorphic to $R^n$

Question

Hier is a result from Topology and Differential Geometry:

A topological manifold is a topological space such that the three conditions are met:

• Hausdorff
• second countable, and
• covered by charts homeomorphic to open subsets in $R^n, \,\,n\in N$.

Statement: None of the three conditions follows from the remaning two. In other words, none of the conditions is dispensible. I can take two of the conditions as holding and eventually suceed in proving analytically the necessity of the third one. What I need is examples which demonstrate that two of the three conditions are not enough in order for a topological space to get the stracture of a topological manifold.

Can somebody show such examples ? Many thanks.

Answer 1

Hausdorf, second countable, not locally homeomorphic to $\mathbb R$:

• $\mathbb Q$.

Hausdorf, not second countable, locally homeomorphic to $\mathbb R$:

• The disjoint union of uncountably many copies of $\mathbb R$.

Not Hausdorf, second countable, locally homeomorphic to $\mathbb R$:

• The line with two origins. As a set, this space is $\mathbb R\cup\{0^*\}$, where $0^*$ is some object not in $\mathbb R$. The open sets consist of all the (usual) open sets in $\mathbb R$, along with those of the form $U\setminus \{0\}\cup \{0^*\}$ and $U\cup\{0^*\}$, where $U$ is any (usual) open subset of $\mathbb R$ containing $0$.