Algebraic definition of a topological space A topological space is a set $X$ with a topology $T \subseteq \mathcal{P}(X)$ satisfying certain axioms involving the operators $\cap, \cup, \subseteq$. $\cap$ takes 2 arguments, $\cup$ takes any number of arguments, $\subseteq$ is a predicate taking 2 arguments.
If we forget about $X$, then $T$ became "just a set", with algebraic operators $\cap, \cup, \subseteq$ on it. Can we still express the fact that it is a topology, using algebraic axioms involving $\cap, \cup, \subseteq$?
 A: As mentioned by Daniel Schepler in the comments you are probably looking for something like the theory of frames and locales. 
Pointless Topology a la wikipedia
Given a topological space $(X,\mathcal{T})$ the collection of open sets $\mathcal{T}$, when ordered by inclusion, makes up what is called complete lattice which is a poset $(L,\leq)$ with the property that every subset $S\subseteq L$ has a supremum, denoted $\bigvee S$ and called the "join" of $S$, as well as an infimum, denoted $\bigwedge S$, called the ``meet" of $S$. In our topology $\mathcal{T}$ we have that $\bigvee\mathcal{T}=X$ and $\bigwedge\mathcal{T}=\emptyset$ the upper and lower bounds for the poset. If you are only considering two open sets $U,V\in\mathcal{T}$ we usually denote the meet and join over the two element set as simply $U\wedge V$ and $U\vee V$, which in the complete lattice of open sets are defined as follows,
$$U\wedge V=U\cap V$$
$$U\vee V=U\cup V$$
EDIT: Hurkyl pointed out my error. Of course, I said that the lattice was complete so arbitrary meets and joins need to be defined. For joins this is easy. If $S$ is an arbitrary collection of open sets then
$$\bigvee_{V\in S}V=\bigcup_{V\in S}V$$
$$\bigwedge_{V\in S}V=int\left(\bigcap_{V\in S}V\right)$$
We needed to take the interior of the intersection because intersections of infinitely many open sets are not always open as you know.
The meet operation doesn't quite use the operations you wanted, but it works well enough.
In addition to being a complete lattice, the topology $\mathcal{T}$ is also satisfies the condition that finite meets distribute over arbitrary joins. That is, if $U\in\mathcal{T}$ and $S\subseteq\mathcal{T}$ is nonempty then
$$U\wedge\left(\bigvee_{V\in S}V\right)=\bigvee_{V\in S}U\wedge V$$
In general a Frame is defined to be a complete lattice satisfying the distributivity condition described above. Why should such a thing be related to topology in general? Well, it turns out that under a fairly weak separation condition one can pass from a topological space to the corresponding frame and then back to the topological space in a canonical way. There are actually two separation conditions that will do the job, but one is called the property of being a sober space.

An element $a\neq 1$ (the upper bounded of a complete lattice) is meet-irreducible if $x\wedge y\leq a$ implies that either $x\leq a$ or $y\leq a$. A topological space is sober if it is $T_{0}$ and the only meet-irreducible elements are those of the form $X\setminus\bar{\{x\}}$. That is the complements of closures of  points.

That definition was shamelessly ripped from:
Frames and Locales
When dealing with the frame associated to a space you abandon the notion of points, but not entirely. There are a few things you can use to serve as a surrogate for points. One such thing is a completely prime filter. Foregoing a verbose description I will refer you to
Ideals and Filters via wikipedia
You replace continuous maps with frame homomorphisms which are monotone maps $f:L\rightarrow K$ that preserve all joins and finite meets. Of course this is backwards from normal continuous functions in which the preimages of open sets are open. To remedy this one often works with the opposite category of locales in which morphisms have the preimage condition.
Frames and Locales are very interesting form a philosophical perspective. For most spaces that people care about the corresponding frame carries all of the information. Likewise the space carries all of the information of the frame. There are frames and locales that do not correspond to any topological space (such frames/locales are called non-spatial), but we can ask ourselves an interesting question. If, when the foundations of topology were being established, our mathematical forefathers had chosen frames instead of point-set spaces would we think of now called non-spatial frames as really being non-spatial? Said differently, what makes a non-spatial frame any less ``spacey" than say, a topological space that isn't even $T_{0}$?
In case you care to read about frames and locales and such shenanigans, the book I use as a reference is:
Frames and Locales
