On page 333 in Section 52: The Fundamental Group (Topology by Munkres) Munkres writes: (see attachement giving Munkres pages 333-334)(adsbygoogle = window.adsbygoogle || []).push({});

"Suppose that [itex] h: X \rightarrow Y [/itex] is a continuous map that carries the point [itex] x_0 [/itex] of X to the point [itex] y_0 [/itex] of Y.

We denote this fact by writing:

[itex] h: ( X, x_0) \rightarrow (Y, y_0) [/itex]

If f is a loop in X based at [itex] x_0 [/itex] , then the composite [itex] h \circ f : I \rightarrow Y [/itex] is a loop in Y based at [itex] y_0 [/itex]"

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 [itex] i^' [/itex] [itex] \in [0, 1][/itex] that is mapped by f into [itex] x^' [/itex] i.e. [itex] f( i^{'} ) [/itex] [itex] = x^' [/itex]

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

i.e. [itex] h \circ f (i^{'} ) [/itex] [itex] = y^' [/itex]

BUT

[itex] h \circ f (i^{'} ) = h(f(i^{'} )) = h(x^{'} ) [/itex]

But (see above) we only know of h that it maps [itex] x_0 [/itex] into [itex] y_0 [/itex]? {seems to me that is not all we need to know about h???}

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

Peter

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# Algebraic Topology - Fundamental Group and the Homomorphism induced by h

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