TITLE: Why $\mathbb{RP}^2$ can not be embedded to $\mathbb{R}^3$?
QUESTION [14 upvotes]: Is there any answer of this question around basic theory of differentiable manifolds?

REPLY [14 votes]: The Stiefel-Whitney class of $\Bbb RP^2$ is $(1+a)^3=a^2+a+1 \in H^*(\Bbb RP^2,\Bbb Z/2)$ where $a$ is the unique nonzero element in $H^1$. The inverse of this element in $H^*(\Bbb RP^2,\Bbb Z/2)$ is $1+a$. As the top Stiefel-Whitney class of the normal bundle of an embedding into $\Bbb R^n$ vanishes (c.f. Milnor-Stasheff Cor. 11.4) $1+a$ is not the Stiefel-Whitney class of the normal bundle to  an embedding into $\Bbb R^3$, hence there is no embedding $\Bbb RP^2\rightarrow \Bbb R^3$. The theory of characteristic classes gives numerous obstructions to embeddings, immersions, cobordisms and other similar geometric constructions. A place to start is the Milnor-Stasheff book Characteristic Classes cited in this answer. 
The above argument easily generalizes to show $\Bbb RP^{2^n}$ does not embed in $\Bbb R^{2^{n+1}-1}$, hence the inequality in the strong Whitney embedding theorem is sharp for dimensions arbitrarily large.