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 Nov 13 answered D^n quotient by its boundary Nov 11 accepted If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree Nov 11 revised If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree edited body Nov 11 comment If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree @MikeMiller For some reason, I wanna do it this way :D, The question is also kinda interesting by itself. Nov 11 comment If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree I'm doing problem 1.2.19 in Hatcher's book. It says if $X=\cup_{n\in Z^+} S^2_n$ where $S^2_n$ is the sphere is center $(1/n,0,0)$ and radius $1/n$, then $X$ is simply connected. Now I'm trying to construct an homotopy of an arbitrary path $f:I\to X$ by pasting some homotopies in $S^2_n$. Nov 11 asked If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree Nov 9 awarded Enthusiast Nov 8 answered Exercise 2 in Hatcher, section 1.2: the union of convex sets is simply connected Nov 8 answered Why is the complement of a discrete subspace of $\mathbb{R}^n$ ($n \ge 3$) simply-connected? Nov 7 comment Union of closed balls centered with centers in a closed set is closed in Euclidean space @NickC Elaborate a little more, please. Nov 6 asked Union of closed balls centered with centers in a closed set is closed in Euclidean space Oct 21 comment Relation between $\langle X\cup Y|R\cup S\rangle$ and $\langle X|R\rangle,\langle Y|S\rangle$ Neat, a proof in the answer would be good though. Oct 21 asked Relation between $\langle X\cup Y|R\cup S\rangle$ and $\langle X|R\rangle,\langle Y|S\rangle$ Oct 20 revised Understanding construction of open nbds in CW complexes added 995 characters in body Oct 18 comment If the derivative is nonzero on an interval, do the endpoints have to be the extrema? Is $f$ defined at $b$? Oct 15 awarded Self-Learner Oct 13 answered A continuous function on a closed subset of real line can be continuously extended Oct 4 accepted Understanding construction of open nbds in CW complexes Oct 3 revised Understanding construction of open nbds in CW complexes added 101 characters in body Oct 3 comment Understanding construction of open nbds in CW complexes The metric is restricted to $D^n-\partial D^n$, but yes, this may lead to misunderstanding so I'll edit the answer accordingly.