Topology MATH 342 Spring 2026

Does mathematics contribute to human flourishing? State your opinion and explain why you hold it. (One-to-three paragraphs expected, but feel free to write more.)

Let \(X\) be a space and let \(Y\) be a space that separates points. Show that the compact-open topology on \(\mathsf{Top}(X,Y)\) separates points.

Let \(\mathbb{R}^{n\times n}\) denote the space of \(n\times n\) real matrices, which is naturally identified with the Euclidean space \(\mathbb{R}^{n^2}\text{.}\) Write \(\mathrm{O}(n)\subseteq\mathbb{R}^{n\times n}\) for the subspace of orthogonal matrices. Consider the function \(\mathbb{R}^n\times\mathbb{R}^{n\times n}\to\mathbb{R}^n\text{,}\) \((x,A)\mapsto Ax\text{.}\) Prove that the restriction of this map to \(\mathbb{R}^n\times\mathrm{O}(n)\to\mathbb{R}^n\) induces an embedding \(\mathrm{O}(n)\hookrightarrow\mathsf{Top}(\mathbb{R}^n,\mathbb{R}^n)\text{,}\) i.e., show that the curried map \(\mathrm{O}(n)\to\mathsf{Top}(\mathbb{R}^n,\mathbb{R}^n)\) is an injective continuous function that is a homeomorphism onto its image. (You will probably want to use the Fundamental Theorem of Topology; please be thorough and careful.)