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The aim of this article is to introduce the notion of a <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>ϕ</mi><mo>,</mo><mi>ψ</mi></mfenced></semantics></math></inline-formula>-metric space, which extends the metric space concept. In these spaces, the symmetry property is preserved. We present a natural topology <inline-formula><math display="inline"><semantics><msub><mi>τ</mi><mfenced separators="" open="(" close=")"><mi>ϕ</mi><mo>,</mo><mi>ψ</mi></mfenced></msub></semantics></math></inline-formula> in such spaces and discuss their topological properties. We also establish the Banach contraction principle in the context of <inline-formula><math display="inline"><semantics><mfenced separators="" open="(" close=")"><mi>ϕ</mi><mo>,</mo><mi>ψ</mi></mfenced></semantics></math></inline-formula>-metric spaces and we illustrate the significance of our main theorem by examples. Ultimately, as applications, the existence of a unique solution of Fredholm type integral equations in one and two dimensions is ensured and an example in support is given.