= Solution
Here is an explicit <simplicial complex> on six vertices. Take every subset of one of the following ten triangular facets:
$$
\boxed{123,\ 124,\ 135,\ 146,\ 156,\ 236,\ 245,\ 256,\ 345,\ 346.}
$$
For example, $135$ denotes $\{1,3,5\}$, with all its edges and vertices included. Its <one-skeleton> has all fifteen edges of the <complete graph>. Every edge belongs to exactly two triangles. The <links of a simplex> at the six vertices are the following cycles, with the first vertex understood to repeat at the end:
$$
\begin{array}{c|c}
\text{vertex}&\text{link cycle}\\\hline
1&2,3,5,6,4\\
2&1,3,6,5,4\\
3&1,2,6,4,5\\
4&1,2,5,3,6\\
5&1,3,4,2,6\\
6&1,4,3,2,5
\end{array}
$$
Thus every vertex has a disk neighbourhood, as do edge and face interior points. The realization is a compact connected <surface> without boundary. Its <Euler characteristic> is
$$
\chi=V-E+F=6-15+10=1.
$$
The <classification theorem for surfaces> identifies the unique closed connected surface with this Euler characteristic as the <real projective plane>. Consequently \b[this complex triangulates $\mathbb{RP}^2$]. The explicit facet and link checks make this a genuine <simplicial complex>, not a polygon whose boundary identifications have been mistaken for simplicial faces.
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