# Hatcher 2.1.28

Hatcher, Algebraic Topology, Chapter 2, Section 1

28. Let $X$ be the cone of the 1-skeleton of $\Delta^3$, the union of all line segments joining the points in the six edges of $\Delta^3$ to the barycenter of $\Delta^3$. Compute the local homology groups $H_n(X,X-\left\{x\right\})$ for all $x\in X$. Define $\partial X$ to be the subspace of points $x$ such that $H_n(X,X-\left\{x\right\})=0$ for all $n$, and compute the local homology groups $H_n(\partial X,\partial X-\left\{x\right\})$. Use these calculations to determine which subsets $A\subset X$ have the property that $f(A)\subset A$ for all homeomorphisms $f:X\to X$.
Proof.

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