$$\textbf{Definition 1}$$ A collection $\mathcal{T}\subseteq\mathcal{P}(X)$ is called a topology on $X$ if it contains $\varnothing$ and $X$, and is closed under unions and finite intersections. $X$ is also called a $\textbf{topological space}$. Any $U\in\mathcal{T}$ is called an $\textbf{open set}$. Any $V$ is closed $\textbf{closed}$ if $X\setminus V$ is open.

$$\textbf{Definition 2}$$ For each $x\in X$, $U\in\mathcal{T}$ is a $\textbf{neighborhood}$ of $x$ if $x\in U$.

$$\textbf{Definition 2}$$ Given $A\subseteq X$. $\\\textbf{1.}$ The $\textbf{closure}$ of $A$ is $\overline{A} = \bigcap\{B\subseteq X\,|\, B\supseteq A, B \text{ is closed}\}$. $\\\textbf{2.}$ The $\textbf{interior}$ of $A$ is $\overset{\circ}{A} = \bigcup\{C\subseteq X\,|\, C\subseteq A, C \text{ is open}\}$. $\\\textbf{3.}$ The $\textbf{exterior}$ of $A$ is $\mathrm{ext}\,A = X\setminus A$. $\\\textbf{4.}$ The $\textbf{boundary}$ of $A$ is $\partial A = X\setminus(\overset{\circ}{A}\cup \mathrm{ext}\,A)$.

$$\textbf{Definition 2}$$ A collection $\mathcal{B}\subseteq\mathcal{P}(X)$ is called a topological basis if $\\ \textbf{1. }$ $X = \bigcup \mathcal{B}$. $\\\textbf{2. }$ $\forall B_1,B_2\in \mathcal{B},\, \forall x \in B_1\cap B_2,\,\exists B_3\in\mathcal{B}$ such that $x\in B_3\subseteq B_1\cap B_2$.

$$\textbf{Theorem 1.1}$$ $\textbf{Find topology from basis.}$ The set $\\\begin{aligned} \mathcal{T}_\mathcal{B} &= \{U \subseteq X \,|\, \forall x\in U,\,\exists B\in\mathcal{B}: x\in B\subseteq U \} \\ &= \left\{\bigcup\mathcal{C} \,|\, \mathcal{C}\subseteq\mathcal{B} \right\} \end{aligned}\\$ is a topology called the topology generated by $\mathcal{B}$.

$$\textbf{Theorem 1.2}$$ $\textbf{Find basis from topology.}$ $\mathcal{C}\in\mathcal{P}(X)$ is a topological basis of $X$ if $\forall x\in U \in\mathcal{T},\,\exists C\in\mathcal{C}$ such that $x\in C \subset U$.

$$\textbf{Theorem 1.3}$$ $\textbf{Compare two topologies.}$ $\\\mathcal{T}_{\mathcal{B}_1}\subseteq\mathcal{T}_{\mathcal{B}_2} \Leftrightarrow \forall x\in B_1\in\mathcal{B}_1,\exists B_2\in\mathcal{B}_2 : x\in B_2\subseteq B_1$.

$$\textbf{Definition 3}$$ An $\textbf{order topology}$ is whose basis contains elements of the form $(a,b), [a_0,b), (a,b_0]$, where $a,b\in X$, $a_0$ is the smallest element if any, $b_0$ is the largest element if any.