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Source of most content: Gunhee Cho and Topology by Munkres. Separation Axiom Def. Given a T.S. $(X,T)$, $X$ is T1 if any finite set is a closed set. Def. Given a T.S. $(X,T)$, $X$ is T2 (Hausdorff) if $\forall x,y \in X$ with $x\ne y$, $\exists$ nbhds $U_x, U_y$ of $x,y$ respectively such that $V_x \cap V_y= \emptyset$. Def. Given a T.S. ...

Source of most content: Gunhee Cho and Topology by Munkres. Countability Axiom Motivation. Let $X$ be a metric space and $A \subseteq X$ be open. Then, If $\exists (x_n) \subseteq A$ s.t. $x_n \to x$, then $x \in \overline{A}$. Note that converse holds if $X$ is a MS. $f$ is cts at $x=p$, i.e., $f(\overline{A}) \subseteq \overline{f(A)}$...

Source of most content: Gunhee Cho, Lecture note by Matsumura and Topology by Munkres. Path-connected Def. Given T.S. $(X, T)$, $X$ is path-connected if $\forall p, q \in X$, $\exists$ cts map $f:[a, b] (\subseteq \mathbb{R}) \to X$ s.t. $f(a)=p$ and $f(q)=b$. We call $f$ a path. NB1; If $X$ is path-connected, then $X$ is connected. Suppos...

Source of most content: Gunhee Cho, Lecture note by Matsumura and Topology by Munkres. Connected Space Def. Given T.S. $(X, T)$, $\emptyset \ne E \subseteq X$ is disconnected if there exist disjoint sets $A, B$ such that (1) non-empty (2) $E \subseteq A \cup B$ (3) $\bar{A} \cap B = \emptyset = A \cap \bar{B}$. NB1; $E$ is connected if $E$ is...

Source of most content: Gunhee Cho, Lecture note by Matsumura and Topology by Munkres. Limit Point, Sequentially Compact Def. Given T.S. $(X, T)$, $X$ is limit point compact if for every infinite subset $A \subseteq X$, $A$ contains a limit point. Def. Given T.S. $(X, T)$, $X$ is sequentially compact if for every seq. $(x _n) _{n=1}^\infty$ ...

Source of most content: Gunhee Cho and Topology by Munkres. Heine-Borel Theorem \[\text{Let } X \text{ be a simply ordered set having the least upper bound property.} \nonumber\] \[\text{Then, any closed interval } [a, b] = \{x\in X : a \leq x \leq b \} \text{ is compact}. \nonumber\] Proof Take any $a,b\in X$ with $a < b$. Let $\m...

Source of most content: Gunhee Cho. Product of Compact Spaces \[\text{Let } X_i \text{ be compact space for } i=1,\ldots, n. \text{ Then, } \prod_{i=1}^n X_i \text{ is compact.} \nonumber\] Tube Lemma \[\text{Suppose } X, Y \text{ be compact. For any } x_0 \in X \text{ and nbhd } U \text{ of } \{x_0 \} \times Y \, \exists \text{ nbhd } W \te...

Source of most content: Gunhee Cho and RCA by Rudin. Convexity Def. For any open set $(a,b) \in [-\infty, \infty]$, $\varphi: (a,b) \to \mathbb{R}$ is convex if $\forall x,y \in (a,b)$ with $x < y$ and $\forall \lambda \in [0, 1]$, $\varphi(\lambda x + (1-\lambda) y) \leq \lambda \varphi(x) + (1- \lambda) \varphi(y)$ holds. NB1; If we pu...