<p>We consider a competitive exchange economy where the commodity space is a locally convex topological vector space ordered by a closed, generating cone. The model includes finitely many consumers, each with a distinct consumption set, assumed to be a closed subcone of the positive cone. Assuming the positive cone has non-empty semi-interior, we establish the existence of equilibrium. This setting extends earlier frameworks that relied on normed and locally convex spaces and shows how equilibrium theory can be developed in more general topological environments. To this end, we develop a new topology on the commodity space, finer than the original one, under which semi-interior points of the positive cone become interior points. This refinement is a key step that enables the extension of equilibrium existence results to locally convex spaces beyond the normed setting. In addition, the analysis proceeds without assuming any lattice structure on the commodity space or a priori continuity assumptions on preferences, thus broadening the applicability of the equilibrium framework. We conclude with an example that illustrates the validity of our results in an infinite-dimensional locally convex setting where standard assumptions such as lattice structure or a priori continuity of preferences are not imposed.</p>