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- Jun 22, 2012

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

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

We denote this fact by writing:

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

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

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

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

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

BUT

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

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

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

Peter

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

We denote this fact by writing:

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

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

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

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

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

BUT

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

But (see above) we only know of h that it maps [tex] x_0 [/tex] into [tex] y_0 [/tex]? {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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