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We propose a systematic definition of genus-$g$, $n$-point, $b$-boundary,$c$-crosscap correlation functions $\mathcal{F}_{g,n,b,c}^{x}$ with $x$boundary operators, for general two-dimensional conformal field theory. The$\mathcal{F}_{g,n,b,c}^{x}$ are defined as the inner products of the surfacestates with the tensor product of $n$ asymptotic states, $b$ boundary statesand $c$ crosscap states, where the $b$ defining boundary states carry a totalof $x$ boundary operators. The definition implies that$\mathcal{F}_{g,n,b,c}^{x}$ are infinite linear combinations of genus-$g$,$(n+b+c)$-point functions $\mathcal{F}_{g,(n+b+c)}$. A single pole structure isidentified in the expansion coefficients $\mathcal{F}_{g,n,b,c}^{0}$, which isanalogous to the single pole structure in the conformal block decomposition.This linear property unifies boundary and bulk CFTs. The consistency conditionsof $\mathcal{F}_{g,n,b,c}^{x}$ are discussed: conventional upper-half plane(UHP) and crosscap constraints can be reformulated as extra constraints on acollection of bulk correlation functions.

Genus-$g$, $n$-point, $b$-boundary, $c$-crosscap correlation functions of two-dimensional conformal field theory: Definition and general properties