Topologies such that every point has a finite number of neighborhouds Are there results known about topologies such that every point is just contained in a finite number of open sets (or neighborhoods)?
 A: Assume that every point is contained in a finite number of open sets
and denote $O_{x}$ for the intersection of open sets containing $x$.
Then $O_{x}$ is open and $N$ is a neighborhood of $x$ iff $O_{x}\subset N$.
If the topology is $T_{1}$ then $O_{x}=\left\{ x\right\} $
for every $x$ and we are dealing with the discrete topology. This on a finite set (if not then you could find infinite open sets that contain $x$). As Svinepels answers (I was just too late) every topology on a finite set will do, simply because in that case the number of open sets is finite. Proved is now that in the non-trivial case (topologies on infinite sets) only topologies that are not $T_{1}$ can meet the conditions.
A: Finite topological spaces would satisfy the property you're asking for. It is known, for instance, that every finite simplicial complex is weakly homotopy equivalent to a finite topological space. This makes it possible to calculate homotopy groups of certain spaces from a finite viewpoint.
A: A class that has been studied are spaces such that each point has a smallest neighbourhood, your spaces are among them.
For such a space denote the smallest neighbourhood containing $x$ by $U_x$. Then $x\le y\iff U_x\subset U_y$ defines a poset (provided the space is $T_0$), and since $U_x=\{y\colon y\le x\}$, the order defines the topology.
Some of the results that Svinepels hints at carry over to this more general setting.
These spaces were first considered by Alexandroff[1] (called discrete spaces, „Der Begriff eines diskreten Raumes ist mit dem Begriff einer teilweise geordneten Menge identisch“) and are now called Alexandroff spaces. Results about homotopy types and weak homotopy types of finite spaces (some of it may either be more general or be carried over to more general settings without problems) are in Stong[2] and McCord[3]. I found these references right now by looking at the list of references in Barmak and Minian[4]. You my also find this webpage by May helpful.
[1] P. Alexandroff, Diskrete Räume, Mat. Sb. (N.S.) 2 (1937), 501-518.
http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=sm&paperid=5579&option_lang=eng
[2] R.E. Stong
Finite topological spaces
Trans. Amer. Math. Soc., 123 (1966), pp. 325–340
http://www.ams.org/journals/tran/1966-123-02/S0002-9947-1966-0195042-2/
[3] M.C. McCord
Singular homology groups and homotopy groups of finite topological spaces
Duke Math. J., 33 (1966), pp. 465–474
[4] Jonathan Ariel Barmak, Elias Gabriel Minian, Simple homotopy types and finite spaces, Advances in Mathematics, Volume 218, Issue 1, 1 May 2008, Pages 87-104, ISSN 0001-8708, http://dx.doi.org/10.1016/j.aim.2007.11.019.
