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This article initiates the study of topological transcendental fields $\FF$ which are subfields of the topological field $\CC$ of all complex numbers such that $\FF$ consists of only rational numbers and a nonempty set of transcendental numbers. $\FF$, with the topology it inherits as a subspace of $\CC$, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is $\QQ(T)$, the extension of the field of rational numbers bydoi:10.48550/arxiv.2202.00837 fatcat:6snt64esdvbplfuaavlk2agtj4