TITLE: Topologies such that every point has a finite number of neighborhouds
QUESTION [7 upvotes]: Are there results known about topologies such that every point is just contained in a finite number of open sets (or neighborhoods)?

REPLY [2 votes]: 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.