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One needs to distinguish between the direct sum and the direct product of a collection of groups. For a countably infinite collection of copies of $\mathbb Z$ the direct sum of these groups is a free …

Lebesgue numbers are certainly not needed to compute $\pi_1(S^1)$ using lifting properties of covering spaces. I checked eleven books that compute $\pi_1(S^1)$ using the covering space ${\mathbb R}\t …

If I understand the question correctly, you have a CW complex $Y'$ which is the union of two subcomplexes $Y$ and $X'$ whose intersection is the subcomplex $X$. We can first collapse $X$ to a point t …

There exist closed orientable hyperbolic 3-manifolds that are surface bundles such that the fiber is the only incompressible surface in the manifold (up to isotopy). Such manifolds can be obtained by …

The quotient group $Diff_1(M)/Diff_0(M)$ is a discrete group since $ Diff_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff …

Maybe the simplest example is the following. There are two fiber bundles with base and fiber both $S^2$. One is the product $S^2\times S^2$ and the other consists of two copies of the mapping cylinder …

The quick argument is to use the fibration $Diff(M,x)\to Diff(M) \to M$ whose total space is the diffeomorphism group of $M$ and whose fiber is the subgroup fixing the point $x$. The projection $Diff …

A counterexample is shown on the cover of the paperback edition of the classic textbook Homology Theory by Hilton and Wylie. This can be viewed on the amazon webpage for the book. The example consis …

(This answer has been edited to give more details.) Finitely generated homotopy groups do not imply finitely generated homology groups. Stallings gave an example of a finitely presented group $G$ suc …

Since the discussion has broadened from the original question to include a wider range of topology books, let me add one more. This is an algebraic topology book by Tammo tom Dieck published just a ye …