Other equivalent definitions of topologies In my book, it defines topological space $\left(X,\mathcal{T}\right)$ satisfying the following axioms:


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*$X$ and $\emptyset$ belong to $\mathcal{T}$

*The union of any number of sets in $\mathcal{T}$ belongs to $\mathcal{T}$

*the intersection of any two sets in $\mathcal{T}$ belongs to $\mathcal{T}$


But there is another definitions of topologies which I don't know why it is equivalent.


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*Let $X$ be a non empty set and let there be assigned to each point $p\in X$ a class $\mathcal{A}_p$ of subsets of $X$ satisfying the Neighborhood Axiom. Then there exists one and only one topology $\mathcal{T}$ on $X$ such that $\mathcal{A}_p$ is the $\mathcal{T}$-neighborhood system of point $p\in X$.

*Let $X$ be a non empty set and let $k$ be an operation which assigns each subset $A$ of $X$ to the subset $A^k$ of $X$, satisfying the Kuratowski Closure Axioms. Then there exists one and only one topology $\mathcal{T}$ on $X$ such that $A^k$ of $X$ is the $\mathcal{T}$-closure of subset $A$ of $X$.

*Let $X$ be a non empty set and let $i$ be an operation which assigns each subset $A$ of $X$ to the subset $i(A)$ of $X$, satisfying
(i) $i(X)=X$, (ii) $i(A)\subset A$, (iii) $i(A\cup B)=i(A)\cup i(B)$, (iv) $i(i(A))=i(A)$
Then there exists one and only one topology $\mathcal{T}$ on $X$ such that $i(A)$ of $X$ is the $\mathcal{T}$-interior of subset $A$ of $X$.
 A: Several resources containing proofs that various methods of generating topology (such as Kuratowski closure operator, neighborhood systems) indeed give a topology have been already mentioned in comments. I am posting a CW answer, where we can collect such references. Feel free to edit this post if you have any relevant addition.
Books

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*R. Engelking in his
book General Topology, in Section 1.2, gives several methods of
generating topology (neighborhood system, closure operator) and he
includes proofs that they indeed give a topology.


*Kelley's General Topology shows when net convergence determines a topology. Kuratowski closure operator is discussed in Chapter 1 (Theorem 8).


*In S. Willard's General topology you can find the result about closure operator in Theorem 3.7 (it is also mentioned in this question) and neighborhood systems are mentioned in Theorem 4.2.


*In Mendelson's Introduction to Topology you can find in Section III.3 the concept of neighborhood space and in Section III.4 the concept of closure space. Again, they are shown to generate a topology. (This book was mentioned in Peter Tamaroff's comment.)
Online

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*Section 3 of Pete L. Clarks notes on general topology notes available from his website.

Blog posts

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*On the The Unapologetic Mathematician's blog you can find definition of topology using bases, neighborhoods and closure.

Searches

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*A lot of results returned when you search for closure operator topology seem to be relevant for your question. You might try the same query in Google Books
