I am trying to find a elementary proof of the Brouwer's fixed point theorem only using basics of point set topology and real analysis. In the one of the textbooks I read, they were proving Brouwer's fixed point theorem for n = 2 the following way:
Let $K \subset \mathbb{R}^2$ be compact and convex. Then consider the map $T:K \to K$, have no fixed points. Now define $F:K \to \partial K$ as follows. For $x \in K$, form a line segment connecting $x$ and $T(x)$, Let $F(x)$ be the point where the line segment intersection the boundary $\partial K$
Now if $T(x) \in \partial K$ then let $F(x)$ equal that intersection point of the ray with $\partial K$ which is not equal to $T(x)$ (The above set up makes sense).
Then they define $F(x) = \dfrac{Tx - x}{\| Tx - x\|}$ (How is this a retraction?), they claim this is a continuous function without proof (what would be the approach of proof here, which definition of continuity should I try?). After this they just say it contradicts the no retraction theorem (result of Sperner's lemma, that's fine).
I would like to get clarification with this proof.
The most elementary analytic proof (as opposed to algebro-geometric) is the one of Milnor-Rogers - Actually, it is Rogers proof, which simplifies the one of Milnor - See proof.
As you don't seem to mind Algebraic Geometry, you can do the proof of Brouwer's Fixed Point Theorem in 2-D just using the fact that the fundamental group of $S^1$ is $\mathbb Z$.