## Random math: 26

Source of most content: Gunhee Cho and Lecture notes by Parimal Parag. Cover and Compact Def. Given a T.S. $(X, T_X)$, $\{ A_\alpha \}_{\alpha \in J} \subset 2^X$ is a cover of $X$ if $X \subset \bigcup _{\alpha \in J} A _\alpha$. If $\{ A _\alpha \} _{\alpha \in J} \subset T_X$, then it is called open cover of $X$. Def. A T.S. $(X, T_X)$ ...

Source of most content: Gunhee Cho. Product Topology Def. Given an index set $J$, let $(A _\alpha) _{\alpha \in J}$ be a seq. of sets. Then, Generalized Cartesian product of $(A _\alpha) _{\alpha \in J}$ is $\Pi _{\alpha \in J} A _\alpha := \{ f: J \rightarrow \cup _{\alpha \in J} A _\alpha : \forall \alpha, f(\alpha) \in A _\alpha \}$. Fo...
Source of most content: Gunhee Cho and Lecture note by Matsumura Hausdorff Space Def. Given T.S. $(X, T_X)$, $\{ a_n\}_{n=1}^\infty \subseteq X$ and $a \in X$. We say $\{ a_n \}$ converges to $a$, $a_n \stackrel{n \rightarrow \infty}{\rightarrow} a$ if $\forall$ nbhd $V$ of $a$, $\exists N \in \mathbb{N}$ s.t. $\forall n \geq N$, $a_n \in V$....