I am trying to understand the (Kuratowski) closure operator (https://en.wikipedia.org/wiki/Kuratowski_closure_axioms). The idea is pretty clear to me however I got stuck in the proofs and maybe this is due to a wrong/unclear definition of the terms I am using:
I defined a topology by a subset $\mathcal T$ of the powerset $\mathcal P(\Omega)$ which includes $\emptyset$, $\Omega$ and is stable under finite intersection/arbitrary unions. Moreover, the interior/closure operator are defined by
$$\mathsf{interior}(V) = \bigcup\Big\{X\in\mathcal T : X\subseteq V\Big\}\\\mathsf{closure}(V) = \bigcap\Big\{\complement X\in\mathcal T : V\subseteq X\Big\}$$
for $V\in$ the powerset. Note that I do not use the terms "open" or "closed" set which is why I have to use $\complement X\in\mathcal T$ to indicate $X$ is closed.
Are these definitions correct? I usually encounter the terms with the terms "open" or "closed set" (e.g. the interior of $V$ is then defined as "the largest open subset of $V$) but as I said these terms are not available for me.
Best Answer
The closure of $V$ is the intersection of all closed subsets of $X$ that contain $V$ as a subset. And a closed set in $X$ is one whose complement relative to $X$ is in the topology $\mathcal{T}$.
So $$\operatorname{cl}(V) = \bigcap \{A \subseteq X: (X\setminus A \in \mathcal{T}) \land (V \subseteq A)\}$$
In the OP's definition of closure he's intersecting the complement of $X$? which is empty, as $X$ is the whole space.