Math and stuff

Proposition If $X_1, \cdots, X_n$ are topological spaces, each canonical projection $\pi_i: X_1 \times \cdots \times X_n \rightarrow X_i$ is continuous. Solution We will use the Characteristic Property of the Product Topology (Theorem 3.27, Introduction to Topological Manifolds). The identity function $f: X_1 \times \cdots \times X_n \rightarrow X_1 \times...

Proposition Suppose \((X_{\alpha})_{\alpha \in A}\) is an indexed family of nonempty $n$-manifolds. Show that the disjoint union \(\coprod_{\alpha \in A} X_{\alpha}\) is an $n$-manifold if and only if $A$ is countable. Solution Suppose $A$ is countable. By the properties of disjoint unions that we know, Since each $X_{\alpha}$ is Hausdorff,...

Proposition Let $M$ be a metric space, and let $S \subset M$ be any subset. Show that the subspace topology on $S$ is the same as the metric topology obtained by restricting the metric of $M$ to pairs of points in $S$. Solution Let $\mathcal{B}_1 = \{ S \cap B_M(x,...

Proposition Let \((X_{\alpha})_{\alpha \in A}\) be an indexed family of topological spaces. A subset of \(\coprod_{\alpha \in A} X_{\alpha}\) is closed if and only if its intersection with each $X_{\alpha}$ is closed. Each canonical injection \(i_{\alpha}: X_{\alpha} \rightarrow \coprod_{\alpha \in A} X_{\alpha}\) is a topological embedding and an open and...

Proposition For each $n \in \mathbb{N}$, $S^n = \{ x \in \mathbb{R}^{n + 1} \mid \abs{x} = 1 \}$ is an $n$-manifold. Solution Let $n \in \mathbb{N}$ be given. We will show that $S^n$ is an $n$-manifold. Since $S^n$ is a subspace of $\mathbb{R}^{n + 1}$, it is Hausdorff and...