Exercise 15, § 5

An espace topologique X is said to possess the Phragmén-Brouwer property if it is connected and if, for every pair $(A,B)$ of disjoint closed subsets of X such that X-A and X-B are connected, then $X-(A\cup B)$ is connected.

a) Let X be a locally connected topological space which possesses the Phragmén-Brouwer property. Let $(A,B)$ be a pair of disjoint closed subsets of X, let $(a, b)$ be a pair of points of $X-(A\cup B)$ which belong to the same connected component of X-A (resp. of X-B). Prove that $a$ and $b$ belong to the same connected component of $X-(A\cup B)$.

b) Let X be a connected and locally path connected topological space which does not possess the Phragmén-Brouwer property. Prove that, for every point $x\in X$, there exists a surjective homomorphism from $\pi_1(X, x)$ onto $\mathbf{Z}$. c) Let X be a connected and locally path connected topological space which is simply connected by arcs. Prove that X possesses the Phragmén-Brouwer property.

d) Let X be a separated and locally connected topological space such that, for every point $x\in X$, the space X$-\{x\}$ possesses the Phragmén-Brouwer property. Let A be a subset of X which is homeomorphic to $[0,1]$; prove that X-A is connected. (Let $c:\mathbf{I}\rightarrow A$ be a homeomorphism. Reason by contradiction and construct a sequence $(J_n)$ where, for every $n,J_n$ is an interval of $\mathbf{I}$ of length $2^{-n}$, contained in $J_{n-1}$ if $n\geqslant 1$, such that X $-c(J_n)$ is not connected; let $x$ be the unique point of X common to all the sets $c(J_n)$. Prove then that X $-\{x\}$ is not connected.)

e) Let $n$ be an integer $\geqslant 1$ and let A be a subset of $\mathbf{S}_n$ which is homeomorphic to $[0,1]$. Prove that $\mathbf{S}_n-$ A is connected.