# Tag Archives: Brouwer’s Fixed Point Theorem

## Around Jordans Curve Theorem I

There are more or less three theorems that are often called Jordan Curve Theorems, while there is a distinction between them. Let us denote by E the Euclidean plane, $$E=\mathbb{R}^2$$ and by $$E^n$$ the Euclidean n-dimensional space, $$E^n=\mathbb{R}^n$$. The Jordan … Continue reading

## Even More On BFPT

I wouldn’t mind writing this unless I haven’t already posted two different proofs of Brouwer’s Fixed Point Theorem. Namely a friend of mine and a reader of this blog, exposed me to another way of proving it. This proof has … Continue reading

Theorem (Brouwer’s Fixed Point Theorem, BFPT). Suppose that $$B\subset \mathbb{R}^n$$ is the closed n-dimensional unit ball and $$f\colon B\to B$$ is a continuous function. Then there exists a point $$x\in B$$ such that $$f(x)=x$$. Theorem (Jordan’s Curve Theorem) Let [tex]f\colon … Continue reading