전체 글182 Homomorphism and Ideal Thm 1Let $\phi : R \to R'$ be ring homomorphism. Then1. $\phi(0) = 0$2. $\forall a \in R$, $\phi(-a) = -\phi(a)$3. $S$ is subring of $R$ $\Rightarrow$ $\phi[S]$ is subring of $R'$4. $S'$ is subring of $R'$ $\Rightarrow$ $\phi^{-1}[S]$ is subring of $R$5. $\phi(1_{R}) = 1_{\phi[R]}$( In general, $\phi(1_{R}) \neq 1_{R'}$ )6. $\ker \phi = \left\{0 \right\}$ $\Leftrightarrow$ $\phi$ is.. 2024. 11. 12. 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. Interior, Boundary, and Exterior Def 1Let $(X, \mathscr{T})$ be a topological space and $A \subseteq X$.1. $x$ is called interior point of $A$$\Leftrightarrow$ $\exists U \in \mathscr{T}$ s.t. $x \in U \subseteq A$ 2. The set of all interior point of $A$ is called interior of $A$.We write $A^{\circ}$ or $\mathrm{Int}(A)$ 3. $x \in X$ is called boundary point of $A$$\Leftrightarrow$ $x \in \overline{A} \cap \overline{.. 2024. 9. 15. 이전 1 2 3 4 ··· 46 다음