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Update theorems/T000881.md
Co-authored-by: Patrick Rabau <70125716+prabau@users.noreply.github.com>
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theorems/T000881.md

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name: Answer to "Cardinality of a vector space versus the cardinality of its basis"
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The vector space $V$ of finite formal linear combinations of elements from $X$ over $\mathbb{R}$ has the
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same cardinality as $X$; see {{mathse:541116}}. Hence there exists a bijection
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$X \to V$. With the indiscrete topology $V$ is a real TVS,
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and this bijection is a homeomorphism
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Suppose $|X|=\kappa\ge\mathfrak c$.
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Let $V$ be a vector space over $\mathbb R$ of (algebraic) dimension $\kappa$.
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As shown in {{mathse:541116}}, $V$ also has cardinality $\kappa$.
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With the indiscrete topology $V$ is a real TVS and is homeomorphic to $X$ via any bijection between the two.

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