Title: Density character of subgroups of topological groups
Abstract: We give a complete characterization of subgroups of separable topological groups. Then we show that the following conditions are equivalent for an <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="omega"> <mml:semantics> <mml:mi>ω</mml:mi> <mml:annotation encoding="application/x-tex">\omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-narrow topological group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>: (i) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homeomorphic to a subspace of a separable regular space; (ii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically isomorphic to a subgroup of a separable topological group; (iii) <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically isomorphic to a closed subgroup of a separable path-connected, locally path-connected topological group. A <italic>pro-Lie group</italic> is a projective limit of finite-dimensional Lie groups. We prove here that an almost connected pro-Lie group is separable if and only if its weight is not greater than the cardinality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German c"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of the continuum. It is deduced from this that an almost connected pro-Lie group is separable if and only if it is homeomorphic to a subspace of a separable Hausdorff space. It is also proved that a locally compact (even feathered) topological group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> which is a subgroup of a separable Hausdorff topological group is separable, but the conclusion is false if it is assumed only that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is homeomorphic to a subspace of a separable Tychonoff space. We show that every precompact (abelian) topological group of weight less than or equal to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German c"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is topologically isomorphic to a closed subgroup of a separable pseudocompact (abelian) group of weight <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German c"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">c</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {c}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This result implies that there is a wealth of closed non-separable subgroups of separable pseudocompact groups. An example is also presented under the Continuum Hypothesis of a separable countably compact abelian group which contains a non-separable closed subgroup.