[Top] Different Definitions of Topology

 Many topology texts introduce general topological spaces with the defintion using open sets and its condition for being an open set. But we can define a topology of a certain set in differen Topological space as a mathematical object is merely a 'pair', $(X,\mathcal{T})$ equipped with a set $X$ and a collection  $\mathcal{T}\subseteq \mathcal{P}(X)$ which we call the 'topology', that satisfies some conditions. Now $\mathcal{T}$ defines the topology, and using open sets, closed sets, neighborhoods, closures, and interiors we can define in different 'words' but all equivalent.

 As I mentioned, most of the topology texts define a topology of set by indtroducing open sets, so defining topology by open sets are familiar to us. We define a 'collection of open sets' of a given  set, $\mathcal{T}\subseteq \mathcal{P}(X)$ and we call the elements of this collection open sets. $\mathcal{T}$ should satisfy, 

$$ \begin{split} \emptyset ,X &\in \mathcal{T} \\ \mathcal{S} \subseteq \mathcal{T} &\Rightarrow \cup \mathcal{S} \in \mathcal{T} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cap V \in \mathcal{T} \end{split}$$

 Now using a similar object known as the closed sets, we can define a topology in a slightly different way. Closed sets are compliments of open sets, so we define a 'collection of closed sets' $\mathcal{C} \subseteq \mathcal{P}(X)$ and the elements of this collection are closed sets. $\mathcal{C}$ should satisfy,

$$ \begin{split} \emptyset ,X &\in \mathcal{C} \\ \mathcal{S} \subseteq \mathcal{C} &\Rightarrow \cap \mathcal{S} \in \mathcal{C} \\ U,V \in \mathcal{T} &\Rightarrow \text{ } U \cup V \in \mathcal{C} \end{split}$$

 The equivalence of the first two definitions seem trivial by the nature of open sets and closed sets. As they are defined by the compliment of each other, it seems trivial. And it also makes $\emptyset$ and $X$ a trivial 'clopen' set. We call the nontrivial clopen sets as proper clopen sets, and whether there exists a proper clopen set is related with the connectedness of the given topological space. ( will be continued in later posts )

  We can also define a topology using neighborhoods. A neighborhood is defined by a mapping $\mathcal{N}:X\rightarrow\mathcal{P}(\mathcal{P}(X))$ which maps $x \in X$ as $x \mapsto \mathcal{N}_{x}$. We call $\mathcal{N}_{x}$ as the neighborhood of $x$ when it satisfies the following conditions.

$$\begin{split} \forall x \in X : N\in \mathcal{N}_{x} &\Rightarrow x \in N \\ N\in\mathcal{N}_{x} \land N\subseteq S \subseteq X &\Rightarrow S\in\mathcal{N}_{x} \\ M,N\in\mathcal{N}_{x} &\Rightarrow M\cap N \in \mathcal{N}_{x} \\ N \in \mathcal{N}_{x} &\Rightarrow \exists\mathcal{N}_{x} \text{ } s.t \text{ } \forall y\in M, N\in\mathcal{N}_{y} ,M\in\mathcal{N}_{x}\end{split}$$

 Now considering closures and interiors, someone may ask "How can we think of closures and interiors when we even don't have the notion of open/closed sets?" The answer to this question is to regard closures and interiors as a mapping, $cl$ and $int$ which makes the pair $(X,cl)$ and $(X,int)$ a topological space. The mappping $cl:\mathcal{P}(X)\rightarrow\mathcal{P}(X)$ is defined with below conditions.

$$\begin{split} cl\text{ }\emptyset &= \emptyset \\ \forall A \subseteq X &\Rightarrow A \subseteq cl \text{ } A \\ \forall A,B\subseteq X &\Rightarrow cl(A\cup B)=cl(A)\cup cl(B) \\ \forall A\subseteq X &\Rightarrow cl(cl\text{ }A)=cl\text{ }A \end{split}$$

 The mapping $int:\mathcal{P}(x)\rightarrow\mathcal{P}(x)$ is defined with below conditions.

$$\begin{split} int\text{ }X &= X \\ \forall A \subseteq X &\Rightarrow A \supseteq int \text{ } A \\ \forall A,B\subseteq X &\Rightarrow int(A\cap B)=int(A)\cap int(B) \\ \forall A\subseteq X &\Rightarrow int(int\text{ }A)=int\text{ }A \end{split}$$

 Citing nLab, the definitions mentioned above can be written again as, a set with a frame of open sets, a set with a co-frame of closed sets satisfying the dual axioms, a set with any collection of subsets, a pair $(X,int)$ where $int$ is a left exact comonad on $\mathcal{P}(X)$, and a pair $(X,cl)$ where $cl$ is a right exact Moore closure operator satisfying the dual of $int$.