Properties
Figure out why all graphs can be embedded in \(\mathbb{R}^3\). Also figure out how Kuratowski managed to find the conditions for a graph to be able to be embedded in \(\mathbb{R}^2\).

Properties
Figure out which real projective spaces are orientable/nonorientable, and why. Also; think about 'minimal non-orientable spaces' w.r.t. CW-complexes...

Properties
Can all manifolds admit CW structure? Apparently its kind of hard to tell (in some sense?) "For compact 4-manifolds the problem of existence of a CW-complex structure is open"? Question 1.3. But "A compact manifold is homotopy equivalent to a CW complex" apparently? So they don't admit CW structure but are homotopy equivalent? That's fine isn't it?

Properties
Suppose we have like a CW-structure on some space \(X\) given by \(K^0,K^1,...\). Is taking the limit (or the inverse limit) of these equivalent to \(X = \cup K^i\)?

Properties
Can all finite CW-complexes be embedded in some finite Euclidean space? If so; what determines how big of a space is necessary to embed a particular CW-complex?