Example of Something That's Not A Manifold

Question

Two examples of non-manifolds that I know are the cross and the cone. Also the sphere with a hair isn't a topological manifold. But what's an example of a topological space $X$ such that $X$ is not a manifold and $X\setminus\{p_1,\ldots p_n\}$ is not a manifold for all $n\in\mathbb{N}$ and $p_i\in X$?

Answer 1

Just to take this off the unanswered list, here are some of the answers from the comments: Feel free to add to this list, or edit it in any way that makes sense.

$\bullet$ The cantor set : Adam Hughes

$\bullet$ If manifolds with boundary are not no be considered manifolds: the closure of any non-empty , non-dense subset of $\mathbb{R}^{n\geq 2}$ :user7530

$\bullet$ Union of a disk and a line: 900 sit-ups a day (that is impressive) (this one would depend one how we are unionizing a disk and a line, for instance, the disjoint union would still be a manifold)

$\bullet$ Or any infinite set with the indiscrete topology, or any set with any topology which is not Hausdorff, et cetera.: Adam Hughes

$\bullet$ A subclass of the last is, Take any Euclidian space, En and an open proper subset, S (such as any open ball) and form the quotient space En/S. Points on the boundary of the open ball will cluster around the point that is is obtained by quotienting by S.: Baby Dragon

$\bullet$ A space that is not locally compact around any point (such as ℚ), a space that's not locally path-connected, a space that's not Hausdorff, etc. : dorebell

$\bullet$ Just put together pieces of different dimensions, sphere with a hair won't be a manifold even if you remove connecting point since pieces aren't locally homeomorphic to the same ℝn. If you don't like that just attach hairs at countably many points or even at a continuum of points.: Conifold

And you can probably find many more examples in the book Couterexamples in Topology.