Math and stuff

Proposition Suppose $X$ is a topological space, and $\mathcal{B}$ is a basis for its topology. Show that a subset $U \subset X$ is open if and only if it satisfies the following conditions: \[\begin{align*} \forall p \in U, \exists B \in \mathcal{B}, p \in B \subset U. \end{align*}\] Solution Suppose...

Proposition Show that a topological space is a $0$-manifold if and only if it is a countable discrete space. Solution Suppose that $X$ is $0$-manifold. Since it is a $0$-manifold, for each $x \in X$, there must exist a neighborhood $U$ that is homeomorphic to an open set in $\mathbb{R}^0$....

Proposition Homotopy equivalence is an equivalence relation on the class of all topological spaces. Solution Let $X, Y, Z$ be given. The identity mapping $i_X$ on $X$ is continuous. Moreover, $i_X \circ i_X = i_X$, so $i_X \circ i_X \simeq i_X$. Therefore, $X \simeq X$. Suppose $X \simeq Y$. Then...

Proposition The quotient topology is indeed a topology. Solution Let $(X, \mathcal{T})$ be a topological space, $Y$ be any set, $q: X \rightarrow Y$ be a surjective map. Let $\mathcal{T}’ = \{ V \subset Y \mid q^{-1}(V) \in \mathcal{T} \}$. We claim that $\mathcal{T}’$ is a topology. $q^{-1}(\emptyset) = \emptyset$,...

Proposition Show that the disjoint union topology is indeed a topology. Solution Let \((X_{\alpha})_{\alpha \in A}\) be an indexed family of nonempty topological spaces. Let \(\mathscr{T}_{\alpha}\) be the topology on $X_{\alpha}$ for each $\alpha \in A$. Let $\mathscr{T}$ denote the disjoint union topology on $\sqcup_{\alpha \in A} X_{\alpha}$. $\forall \alpha...