Components & Inclusions
due by Monday, Sep 11, 2023

Let
$X$
be a topological space and$x_0$
be a point in$X$
. Let$Z$
be the pathcomponent of$x_0$
in$X$
. Show that the inclusion$i: Z \hookrightarrow X$
induces an isomorphism on fundamental groups. 
Let
$f: (X, x_0)\to (Y, y_0)$
be a map between based topological spaces and let$f_*: \pi_1(X, x_0)\to \pi_1(Y, y_0)$
be the induced map on fundamental groups. Prove or disprove the following statements: (a) If
$f$
is injective then so is$f_*$
.  (b) If
$f$
is surjective then so is$f_*$
.  (c) If
$f$
is a homeomorphism then$f_*$
is an isomorphism.
 (a) If