Lemma: If $X$ is a CW complex then:
a) $H_{k}(X^{n},X^{n-1})$ is zero for $k\neq n$ and is free abelian for $k=n$ with a basis in one-to-one correspondence with the n-cell of $X$.
b) $H_{k}(X^{n})=0$ for $k>n$. In particular, if $X$ is finite-dimensional $H_{k}(X)=0$ for $k> Dim(X)$.
c) The map $H_{k}(X^{n})\longrightarrow H_{k}(X)$ induced by the inclusion $X^{n}\hookrightarrow X$ is an isomorphism for $k<n$ and surjective for k=n.
my problem is how to show part c) for the infinite-dimensional case, that is, when X is of infinte-dimensional. If someone can give me a clue how to do it, I will thank you very much.