I want to relate the homology of the suspension and the cone of a space by proving they are equal: $\widetilde{H}_n(CX,X)\cong \widetilde{H}_n(SX) $.
Here $ CX$ is the cylinder with all the base point identified: $CX= X\times I/{(x,s)\sim(x,y)\leftrightarrow s=t=1 } $. And $SX $ is the union of two cones identifying the bases. Equivalentley, $ SX$ is the cylinder of $ X$ with all the base points identified and all the ceiling points idenfitied: $SX=X\times I/{(x,s)\sim(x,y)\leftrightarrow s=t=1 \; \text{or}\; s=t=1} $