Problem Sheet I

This sheet is based on Lectures 17 and 18.

• Problems 1 and 2 are famous results of Brouwer. Problem 2 implies that $ \mathbb{R}^m $ is not homeomorphic to $ \mathbb{R}^n$ if $n \ne m$ (and thus the concept of "dimension" makes sense in algebraic topology.) Hint: Use the Jordan-Brouwer Separation Theorem.
• Problems 3 is about point-set topological properties of weakly Hausdorff spaces.
• Problem 4 asks you to prove that adjunction spaces are pushouts in $ \mathsf{Top}$, together with other useful properties.
• Problem 5 and 6 are examples of how to compute the homology of a space constructed by attaching cells.

As usual, feel free to ask questions if you are stuck! 😀