An elementary proof of invariance of domain and its consequences

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bachelor_Wu_Yiyou_2026.pdf (617.43 KB)

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School of Engineering | Bachelor's thesis
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Date

2026-02-24

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Computational Engineering

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Degree programme

Aalto Bachelor's Programme in Science and Technology
Aalto Bachelor's Programme in Science and Technology

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en

Pages

25

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Abstract

Let be a open subset of a Euclidean space R. Then, given any continuous injective mapping from to R, the image () is also open in R. The above is the statement of the invariance of domain theorem for Euclidean space; In this thesis, we present an elementary proof of the theorem that does not involve tools from algebraic topology and introduce an important consequence of the theorem: Euclidean spaces of different dimensions are not homeomorphic.

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Supervisor

St-Pierre, Luc

Thesis advisor

Koski, Aleksis

Keywords

topology, invariance of domain, Brouwer’s fixed point theorem

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https://urn.fi/URN:NBN:fi:aalto-202603032646

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