For 2)\Rightarrow 1). Let F:A\times I\hookrightarrow X\times 0\cup A\times I be the inclusion which defines a homotopy starting from the restriction of i_0:X\cong X\times 0\hookrightarrow X\times 0\cup A\times I ...

Building on what you have already proved for irrational points: Assume that a rational point b of \overline{B} is not in B. Then E_b is not in A. Since b is in \overline{B}, there is a ...

If you take p to be the quotient map p:(M(f),X \times \{0\})\to(C(f),P) (where P is the point X \times \{0 \} is collapsed to) the map this p induces in homology is what I believe you are ...

To show that u \circ \pi_1 = id consider that u \circ \pi_1 is a morphism from X\times 1 \to X\times 1 such that \pi_1 \circ u \circ \pi_1 = \pi_1 and \top_{X\times 1} = \top_{X\times 1} \circ u \circ \pi_1 ...

A graduation cap is an example of a mapping cylinder g : X \to Y where X = S^1, Y = [-2,2] \times [-2,2], and g is the inclusion map.

To start, it is natural to me (though unnecessary) to work with continuous limits instead of sequences. You should convince yourself that the dominated convergence theorem applies to this case as well ...