Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Connected door spaces and topological solutions of equations   The connected door space is an enigmatic topological space in which every
proper nonempty subset is either open or closed, but not both. This paper
provides an elementary proof of the classification theorem of connected door
spaces. More importantly, we show that connected door topologies can be viewed
as solutions of the valuation $f(A)+f(B)=f(A\cup B)+f(A\cap B)$ and the
equation $f(A)+f(B)=f(A\cup B)$, respectively. In addition, some special
solutions, which can be regarded as a union of connected door spaces, are
provided.

25
The categories of this paper are: patt-sol
34
The categories of this paper are: chem-ph
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph
27
The categories of this paper are: comp-gas
22
The categories of this paper are: chao-dyn
31
The categories of this paper are: ao-sci
15
The categories of this paper are: nucl-ex
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
10
The categories of this paper are: q-bio
6
The categories of this paper are: astro-ph
37
The categories of this paper are: supr-con
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
16
The categories of this paper are: stat