Collapsing either one of the circles in the bouquet of two circles to the basepoint, how can I describe this by a map (in the free product) and how is this related to that $\mathbb{R}^2\backslash\{p,q\}$ is not homeomorphic to a bouquet of two circles! However there is a deformation retraction of that space to a subspace homeomorphic to the bouquet of two circles?
Any help will be greatly appreciated!
If you try to write down the homomorphism induced by the collapsing map from the figure eight to the collapsed space which is a circle, it would be enough to write down the images of the two generators of the fundamental group of the figure eight which is a free group of over 2 generators. Then the homomorphism takes one of the generators to 0 (which corresponds to the collapsed circle) and another one to identity (which corresponds to the intact circle).
For the 2nd question it would be enough to point out that wedge of 2 circles is compact whereas plane without 2 points is not.