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일반위상/위상공간6

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.
limit point Def 1Let $(X, \mathscr{T})$ be a topological space and $A \subseteq X$.$x \in X$ is called limit point (or accumulation point) of $A$ in $X$ if $\forall U \in \mathscr{T}$ containing $x$,  $(U \setminus \left\{x \right\}) \cap A \neq \varnothing$ So $x$ is not limit point of $A$ if there exists $U \in \mathscr{T}$ containing $x$  s.t.$(U \setminus \left\{x \right\}) \cap A = \varnothing$   Prop .. 2024. 9. 15.
Hausdorff Def 1Let $(X, \mathscr{T})$ be a topological space. $X$ is Hausdorff if$\forall a, b \; (a \neq b) \in X$,  there exists $U, V \in \mathscr{T}$  s.t. $a \in U$,     $b \in V$,     $U \cap V = \varnothing$   Thm 1 $X$ is metrizable  $\Rightarrow$  $X$ is Hausdorff 더보기  Let $\mathscr{T}$ be a topology of $X$  $\forall a, b \; (a \neq b) \in X$   Let $r = d(a, b)$  $(>0)$  $B_{\frac{r}{3}}(a), B_{\.. 2024. 9. 15.
Metric space Def 1Let $X$ be a set and $d : X \times X \to \mathbb{R}$ be a function that satisfies the following $3$ conditions. $\forall x, y, z \in X$$(\mathrm{i})$  $d(x, y) \geq 0$     &     $d(x, y) = 0$  $\Leftrightarrow$  $x = y$$(\mathrm{ii})$  $d(x, y) = d(y, x)$ $(\mathrm{iii})$  $d(x, z) \leq d(x, y) + d(y, z)$ $d$ : metric (or distance) on $X$.$(X, d)$ : metric space.     Example1.  Euclidean me.. 2024. 3. 20.
Base Def 1$(X, \mathscr{T})$ : topological spaceA family $\mathscr{B}$ of open sets of $X$ ( i.e. $\mathscr{B} \subseteq \mathscr{T}$ ) is a base for $\mathscr{T}$ ifevery open set in $X$ is a union of sets in $\mathscr{B}$.   Thm 1Let $(X, \mathscr{T})$ be a topological space  and  $\mathscr{C} \subseteq \mathscr{T}$ .T.F.A.E.(1)  $\mathscr{C}$ is base for $\mathscr{T}$  (2)  $\forall p \in X$  and .. 2024. 3. 11.
Topological space Def 1Let $X$ be a set. A family $\mathscr{T}$ of $X$ is a topology of $X$ if $\mathscr{T}$ has the following $3$ properties.  $(\mathrm{i})$   $\varnothing, X \in \mathscr{T}$ $(\mathrm{ii})$   $U_{\alpha} \in \mathscr{T}$  for $\alpha \in I$  $\Rightarrow$  $\displaystyle \bigcup_{\alpha \in I} U_{\alpha} \in \mathscr{T}$$(\mathrm{iii})$   $U_{1}, \cdots, U_{n} \in \mathscr{T}$  $\Rightarrow$  .. 2024. 3. 7.

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