일반위상/Compact Space3 Compact Theorem 2 Thm 1 Bolzano - Weierstrass TheoremLet $(X, \mathscr{T})$ be a compact space and $A$ be an infinite subset in $X$. Then $A$ has a limit point in $X$. 더보기 Suppose : $A$ don't have a limit point in $X$ So for each $x \in X$, there exists $U_{x} \in \mathscr{T}$ containing $x$ s.t. $(U_{x} \setminus \left\{x \right\}) \cap A = \varnothing$ $\therefore$ $\left\{U_{x} \; | \; x \in X \right\}.. 2024. 9. 23. Compact Theorem 1 Thm 1Let $(X, \mathscr{T})$, $(Y, \mathscr{T}')$ be topological space and $A \subseteq X$.Let $f : X \to Y$ be continuous and $A$ be compact. Then$f(A)$ is compact 더보기 Let $\left\{U_{\alpha} \in \mathscr{T}' \; | \; \alpha \in I \right\}$ be open cover of $f(A)$. So $f(A) \subseteq \displaystyle \bigcup_{\alpha \in I} U_{\alpha} $ Since $f$ is continuous, $f^{-1}(U_{\alpha}) \in \mathsc.. 2024. 9. 16. Compact space Def 1Let $(X, \mathscr{T})$ be a topological space.Let $\left\{U_{\alpha} \in \mathscr{T} \; | \; \alpha \in I \right\}$ be a collection of open sets such that$X = \displaystyle \bigcup_{\alpha \in I} U_{\alpha}$ $\left\{U_{\alpha} \in \mathscr{T} \; | \; \alpha \in I \right\}$ : open cover of $X$ Let $\left\{U_{\beta} \in \mathscr{T} \; | \; \beta \in J \right\}$ be subset of $\left\{U_{\alpha.. 2024. 6. 4. 이전 1 다음