There exist a large number of, sometimes surprising, applications of Baire category theorem. For instance, in this blog, we saw that

  • two generic orientation-preserving homeomorphisms of the circle \mathbb{S}^1 generate a non abelian free subgroup,
  • a generic continuous function [0,1] \to \mathbb{R} is nowhere differentiable.

See the previous notes Free groups acting on the circle and Almost all continuous functions are nowhere differentiable. Now, we want to prove:

Theorem: If n \geq 3, the space \mathbb{R}^n \backslash \mathbb{Q}^n is simply connected.

Fix a base point x \in \mathbb{R}^n \backslash \mathbb{Q}^n and an enumeration \mathbb{Q}^n = \{ q_1, q_2, \ldots \}. Given two closed curves c_0,c_1 : [0,1] \to \mathbb{R}^n \backslash \mathbb{Q}^n satisfying c_0(0)=c_1(0)=x, we want to prove that c_0 and c_1 are homotopic (relatively to x) in \mathbb{R}^n \backslash \mathbb{Q}^n. Let

\mathcal{H}(c_0,c_1) = \{ \text{homotopy between} \ c_0 \ \text{and} \ c_1 \ \text{in} \ \mathbb{R}^n \}.

It is a closed subset of the Banach space \left( \mathcal{C}^0([0,1]^2, \mathbb{R}^n), \| \cdot \|_{\infty} \right) of the continuous functions [0,1]^2 \to \mathbb{R}^n with respect to the supremum norm \| \cdot \|_{\infty}, so that \mathcal{H}(c_0,c_1), endowed with the distance induced by \| \cdot \|_{\infty}, is a complete metric space. Now, let

\mathcal{H}^k(c_0,c_1) = \{ h \in \mathcal{H}(c_0,c_1) \mid \mathrm{Im}(h) \cap \{ q_1, \ldots, q_k \}= \emptyset \}.

Because n \geq 3, the space \mathbb{R}^n \backslash \{ q_1, \ldots, q_k \} is simply connected, so that \mathcal{H}^k(c_0,c_1) \neq \emptyset. In fact, it is not difficult to convince ourself that \mathcal{H}^k(c_0,c_1) is a dense open subset of \mathcal{H}(c_0,c_1). A possible argument is the following:

First, if h \in \mathcal{H}^k(c_0,c_1) and if d denotes the distance between \{q_1, \ldots, q_k \} and \mathrm{Im}(h) (which is positive since \mathrm{Im}(h) is compact), then the ball in \mathcal{H}(c_0,c_1) of radius d/2 and centered at h is clearly included into \mathcal{H}^k(c_0,c_1). Therefore, \mathcal{H}^k(c_0,c_1) is open in \mathcal{H}(c_0,c_1).

Then, let f_{\epsilon} be a bump function whose support is included into \bigcup\limits_{j=1}^k B(q_i,\epsilon) and satisfying \| f_{\epsilon} \|_{\infty} \leq \epsilon. Now, given a homotopy h \in \mathcal{H}(c_0,c_1), define the perturbation

h_{\epsilon} : (s,t) \mapsto (t(t-1)f_{\epsilon} + \mathrm{Id}_X) \circ h(s,t).

Of course, \| h-h_{\epsilon} \|_{\infty} \leq \| f_{\epsilon} \|_{\infty} \leq \epsilon and h_{\epsilon} \in \mathcal{H}(c_0,c_1). Furthermore, if \epsilon is small enough and f_{\epsilon} well chosen, then \mathrm{Im}(h_{\epsilon}) \cap \{ q_1, \ldots, q_k \} = \emptyset. This proves that \mathcal{H}^k(c_0,c_1) is dense in \mathcal{H}(c_0,c_1).

Now, it is sufficient to apply Baire category theorem to conclude that the intersection \bigcap\limits_{k \geq 1} \mathcal{H}^k (c_0,c_1) is non empty. But any element h of this intersection defines a homotopy between c_0 and c_1 satisfying \mathrm{Im}(h) \cap \mathbb{Q}^n = \emptyset, that is to say a homotopy in \mathbb{R}^n \backslash \mathbb{Q}^n.

Therefore, we have proved that any two closed curves in \mathbb{R}^n \backslash \mathbb{Q}^n based at x are homotopic in \mathbb{R}^n \backslash \mathbb{Q}^n. This shows that the group \pi_1(\mathbb{R}^n \backslash \mathbb{Q}^n,x) is trivial, ie., the space \mathbb{R}^n \backslash \mathbb{Q}^n is simply connected.

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