From Euclidean Spaces to Metric Spaces


  • Ryan Joseph Rogers University of Kentucky University of Cincinnati
  • Ning Zhong University of Cincinnati


Euclidean Spaces, Metric Spaces


In this note, we provide the definition of a metric space and establish that, while all Euclidean spaces are metric spaces, not all metric spaces are Euclidean spaces. It is then natural and interesting to ask which theorems that hold in Euclidean spaces can be extended to general metric spaces and which ones cannot be extended. We survey this topic by considering six well-known theorems which hold in Euclidean spaces and rigorously exploring their validities in general metric spaces.


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Author Biographies

Ryan Joseph Rogers, University of Kentucky University of Cincinnati

Ph.D., University of Kentucky, Lexington, Kentucky, Expected 2024 (Mathematics).


B.S. (Summa Cum Laude), University of Cincinnati, Cincinnati, Ohio, 2017 (Mathematics).

Ning Zhong, University of Cincinnati

Mathematical Sciences



How to Cite

Rogers, R. J., & Zhong, N. (2019). From Euclidean Spaces to Metric Spaces. Journal of Student Research, 8(1). Retrieved from



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