On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachment giving Munkres pages 333-334)

"Suppose that $\displaystyle h: X \rightarrow Y $ is a continuous map that carries the point $\displaystyle x_0 $ of X to the point $\displaystyle y_0 $ of Y.

We denote this fact by writing:

$\displaystyle h: ( X, x_0) \rightarrow (Y, y_0) $

If f is a loop in X based at $\displaystyle x_0 $ , then the composite $\displaystyle h \circ f : I \rightarrow Y $ is a loop in Y based at $\displaystyle y_0 $"

I am confused as to how this works ... can someone help with the formal mechanics of this.

To illustrate my confusion, consider the following ( see my diagram and text in atttachment "Diagram ..." )

Consider a point $\displaystyle i^' $ $\displaystyle \in [0, 1]$ that is mapped by f into $\displaystyle x^' $ i.e. $\displaystyle f( i^{'} ) $ $\displaystyle = x^' $

Then we would imagine that $\displaystyle i^' $ is mapped by $\displaystyle h \circ f $ into some corresponding point $\displaystyle y^' $ ( see my diagram and text in atttachment "Diagram ..." )

i.e. $\displaystyle h \circ f (i^{'} ) $ $\displaystyle = y^' $

BUT

$\displaystyle h \circ f (i^{'} ) = h(f(i^{'} )) = h(x^{'} ) $

But (see above) we only know of h that it maps $\displaystyle x_0 $ into $\displaystyle y_0 $? {seems to me that is not all we need to know about h???}

Can anyone please clarify this situation - preferably formally and explicitly?

Peter