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Topological band theory provides a conceptual framework to predict or even engineer robust metallic states at the boundaries of topologically distinct phases. The bulk-boundary correspondence requires that a topological electronic phase transition between two insulators must proceed via closing of the electronic gap. Therefore, it can provide a conceptual solution to the instability of metallic phases in {\pi}-conjugated 1D polymers. In this work we predict and demonstrate that a clever design and on-surface synthesis of polymers consisting of 1D linearly bridged polyacene moieties, can position the resulting polymer near the topological transition from a trivial to a non-trivial quantum phase featuring a very narrow bandgap with in-gap zero-energy edge-states at the topologically non-trivial phase. We also reveal the fundamental connection between topological classes and electronic transformation of 1D {\pi}-conjugated polymers.