Example

Let Y be a locally finite-dimensional differentiable manifold and let Z be a closed submanifold of Y (VAR, R, 5.8.3). Put X = Y-Z.

a) If the codimension of Z is at least 2 at every point, the canonical injection of X into Y is relatively connected. Let indeed $z$ be a point of Z; there exists an open neighbourhood V of $z$ in Y and a homeomorphism $\varphi$ of V onto a finite-dimensional vector space E over $\mathbf{R}$ such that $\varphi (V\cap Z)$ is a vector subspace F of E whose codimension is $\geqslant 2$. The set E-F is connected (TG, VI, p. 5, proposition 4), whence the assertion.

b) Suppose moreover that the codimension of Z is at least equal to 3 at every point; the hypotheses of proposition 6 and of its corollary are then satisfied because, with the notations of the preceding paragraph, E-F is simply connected (I, p. 128, example 4).

Differentiable manifolds being contractible spaces (IV, p. 347), the results of this No. admit the following particular case.

Let Y be a locally finite-dimensional differentiable manifold, Z a closed submanifold of Y and $i$ the canonical injection of Y-Z into Y.

a) If the codimension of Z in Y is $\geqslant 1$ at every point, the manifold Y-Z is dense in Y and the mapping $\pi_0(i)$ is surjective.

b) If the codimension of Z in Y is $\geqslant 2$ at every point, the mapping $\pi_0(i)$ is bijective and, for every point $x$ of Y-Z, the mapping $\pi_1(i, x)$ is surjective.

c) If the codimension of Z in Y is $\geqslant 3$ at every point, the mappings $\pi_0(i)$ and $\pi_1(i, x)$ are bijective, for every point $x$ of Y-Z.