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Suppose $V$ is a real vector space of dimension $n$ and let $\varphi: V \to \mathbb{R}^n$ be an isomorphism. Then we can define a topology on $V$ through $\varphi$ by defining $U$ to be open in $V$ iff $\varphi(U)$ is open in $\mathbb{R}^n$.

My question: Given another isomorphism $\psi:V \to \mathbb{R}^n$, is the induced topology by $\psi$ related to the induced topology by $\varphi$? In general there are many such isomorphisms and when there is no canonical choice and these give severly different topologies, then how does one proceed?

Example: If $M$ is a smooth $n$-dimensional manifold the tangent space $T_pM$ is a topological space as a subspace of $TM$ and by $T_pM \cong \mathbb{R}^n$, where an isomorphism is given by choosing a smooth chart. However, every smooth chart induces a different isomorphism and I don't know if the induced topologies are related in any way or not.

user3118
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    Any two Hausdorff vector space topologies on a finite dimensional space coincide. This is a theorem of Tychonov from 1935. It can be found in many text books on functional analysis, e.g., in Rudin's book. – Jochen Jan 28 '23 at 13:50
  • @Jochen Thanks, I didn't know about this. If you rewrite your comment as an answer I will gladly accept it. – user3118 Jan 28 '23 at 13:55

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Any two Hausdorff vector space topologies on a finite dimensional space coincide. This is a theorem of Tychonov from 1935. It can be found in many text books on functional analysis, e.g., in Rudin's book.

Jochen
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