Proposition 7

Let X be a topological space. Let $U_1$ and $U_2$ be connected open ( resp. closed) subspaces of X such that $X = U_1\cup$ $U_2$. If the space X is simply connected, then $U_1\cap U_2$ is connected.

Put $U = U_1\cap U_2$ and suppose, arguing by contradiction, that the space U is not connected. We shall construct a connected covering of X of infinite degree; such a covering is not trivializable.

By assumption, the set U is the union of two disjoint, non-empty sets A and B, open (resp. closed) in X. For $i\in \{1,2\}$, let $Y_i$ be the $U_i$-space $(U_i\times \mathbf{Z}$, pr$_1)$ and let $Z_i$ be the subspace $U\times \mathbf{Z}$ of $Y_i$. The mapping $h: Z_1\rightarrow Z_2$ defined by

$(x, n)$ if $x\in A$ and $n\in \mathbf{Z}$

$(x, n+ 1)$ if $x\in B$ and $n\in \mathbf{Z}$

is a homeomorphism. Let Y be the space obtained by gluing $Y_1$ and $Y_2$ along $Z_1$ and $Z_2$ by means of the homeomorphism $h$ (TG, I, p. 17).

For $i\in \{1,2\}$, the canonical mapping $g_i$ of $Y_i$ into Y is a homeomorphism of $Y_i$ onto an open (resp. closed) subset of Y (loc. cit., prop. 9). For every integer $n\in \mathbf{Z}$, the sets $g_i(U_i\times \{n\})$, $i\in \{1,2\}$, are connected; since A and B are non-empty, $g_1(U_1\times \{n\})$ meets $g_2(U_2\times \{n\})$ and $g_2(U_2\times \{n+1\})$. It follows that the space Y is connected (TG, I, p. 81, corollary).

For $x\in U$ and $n\in \mathbf{Z}$, one has (pr$_1\circ h$)$(x, n) =x=$ pr$_1(x, n)$. There exists therefore a unique continuous mapping $p: Y\rightarrow X$ such that $p\circ g_i=$ pr$_1$ for $i\in \{1,2\}$. Let us prove that the X-space $(Y, p)$ is a covering. By construction, the fibres of the mapping $p$ are homeomorphic to the discrete space $\mathbf{Z}$.

For $i\in \{1,2\}$, the mapping $g_i$ defines by passing to subspaces an isomorphism of $U_i$-spaces of $U_i\times \mathbf{Z}$ onto $\overset{-1}{p}(U_i)$.

By definition of the space Y, there exists a unique mapping $k$ of $(X-A)\times \mathbf{Z}$ into Y such that $k(x, n) =g_1(x, n)$ for $x\in (U_1-A)\times \mathbf{Z}$ and $k(x, n) =g_2(x, n-1)$ for $x\in (U_2-A)\times \mathbf{Z}$; it is an isomorphism of (X-A)-spaces of $(X-A)\times \mathbf{Z}$ onto $\overset{-1}{p}(X-A)$. Analogously, there exists a unique mapping $k'$ of $(X-B)\times \mathbf{Z}$ into Y which coincides with $g_1$ on $(U_1-B)\times \mathbf{Z}$ and with $g_2$ on $(U_2-B)\times \mathbf{Z}$ and it is an isomorphism of (X-B)-spaces of $(X-B)\times \mathbf{Z}$ onto $\overset{-1}{p}(X-B)$.

This proves that the X-space $(Y, p)$ is trivializable over the subsets $U_1,U_2$, X-A and X-B. One has $U_1\cup U_2= X$; if $U_1$ and $U_2$ are open in X, this proves that $(Y, p)$ is a covering of X. The same holds when $U_1$ and $U_2$ are closed in X, for then X-A and X-B are open subsets of X whose union is X. The proposition is thus proved.