Basic Algebraic Topology

Hints and Solutions

• §1.1.19 (page no. 11)
• (i). If H is the homotopy from the identity map to a constant map at x₀, then t ↦ H (x, t) gives a path from x to x₀.
• (ii). There are many such topological properties though giving examples may not be easy. Here are six of them for which the examples are easy.
• (a) compactness; (b) being locally Euclidean of the same dimension; (c) locally Euclidean; (d) locally path connected; (e) locally connectedness; (f) Hausdorffness. For (a) and (b) you can use the fact that ℝⁿ is homotopy equivalent to a single point for all n. For (c), you can use the union of two axes which is contractible. For (d) and (e) you can use the comb space which is not locally connected but contractible. For (f) take the example given ...