| Summary: | This article initiates the study of topological transcendental fields <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> which are subfields of the topological field <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula> of all complex numbers such that <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula> only consists of rational numbers and a nonempty set of transcendental numbers. <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">F</mi></semantics></math></inline-formula>, with the topology it inherits as a subspace of <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">C</mi></semantics></math></inline-formula>, is a topological field. Each topological transcendental field is a separable metrizable zero-dimensional space and algebraically is <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>, the extension of the field of rational numbers by a set <i>T</i> of transcendental numbers. It is proven that there exist precisely <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mn>2</mn><msub><mo>ℵ</mo><mn>0</mn></msub></msup></semantics></math></inline-formula> countably infinite topological transcendental fields and each is homeomorphic to the space <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi mathvariant="double-struck">Q</mi></semantics></math></inline-formula> of rational numbers with its usual topology. It is also shown that there is a class of <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><msup><mn>2</mn><msup><mn>2</mn><msub><mo>ℵ</mo><mn>0</mn></msub></msup></msup></semantics></math></inline-formula> of topological transcendental fields of the form <inline-formula><math display="inline" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula> with <i>T</i> a set of Liouville numbers, no two of which are homeomorphic.
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