Lowlying excited states in a one-dimensional system which has a small energy gap (εg) in the spectra of H0, the one-electron Hamiltonian, are investigated. Plasmons and excitons are our main interests. These states are investigated in the framework of the tight-binding approximation. In usual one-dimensional systems, such as very long linear conjuated molecules, the plasmon levels sink into the level continuum given by excitation energies of one-pair states. However, it is shown that the plasma oscillations are stable when their energies are sufficiently larger than the energy gap. On the other hand, the plasma oscillations with sufficiently small momentum whose energies in case of εg = 0 were smaller than the present energy gap seem to be dissolved away into the level continuum. A formulation to derive exciton solutions is given. The screening effect for the attractive force between the electron and the hole is investigated by means of the Gell-Mann and Brueckner techniqe and is found to grow larger as the energy gap becomes smaller. By this screening effect the possibility of getting the exciton-like bound state diminishies as εg → 0. As a collorary we have found that the potential energy of two electric charges z1e and z2e separated by a distance r in a three dimensional system with small energy gap becomes z1z2e2 exp(-αr)/r if r ≪ |B/ΔB|, where |B| is a measure of the Fermi energy and ΔB is a measure of the gap, while it becomes z1z2e2/εr if r ≫ |B/ΔB|, where ε∝ rs-1(|B|/(ΔB)2) × (e2/aB).