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Articles 1 - 8 of 8

Full-Text Articles in Education

Connectedness- Its Evolution And Applications, Nicholas A. Scoville Apr 2019

Connectedness- Its Evolution And Applications, Nicholas A. Scoville

Topology

No abstract provided.


From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville Jul 2018

From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville

Topology

No abstract provided.


Nearness Without Distance, Nicholas A. Scoville Jul 2018

Nearness Without Distance, Nicholas A. Scoville

Topology

No abstract provided.


The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville Jul 2017

The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville

Topology

No abstract provided.


A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville Jul 2017

A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville

Topology

In a short paper published just one year prior to his thesis, Maurice Frechet gives a simple generalization one what we might today call the Extreme value theorem. This generalization is a simple matter of coming up with ``the right" definitions in order to make this work. In this mini PSP, we work through Frechet's entire 1.5 page paper to give an extreme value theorem in more general topological spaces, ones which, to use Frechet's newly coined term, are compact.


Topology From Analysis, Nicholas A. Scoville Apr 2016

Topology From Analysis, Nicholas A. Scoville

Topology

Topology is often described as having no notion of distance, but a notion of nearness. How can such a thing be possible? Isn't this just a distinction without a difference? In this project, we will discover the notion of nearness without distance by studying the work of Georg Cantor and a problem he was investigating involving Fourier series. We will see that it is the relationship of points to each other, and not their distances per se, that is a proper view. We will see the roots of topology organically springing from analysis.


Connecting Connectedness, Nicholas A. Scoville Apr 2016

Connecting Connectedness, Nicholas A. Scoville

Topology

No abstract provided.


The Cantor Set Before Cantor, Nicholas A. Scoville Apr 2016

The Cantor Set Before Cantor, Nicholas A. Scoville

Topology

A special construction used in both analysis and topology today is known as the Cantor set. Cantor used this set in a paper in the 1880s. Yet it appeared as early as 1875 in a paper by the Irish mathematician Henry John Stephen Smith (1826 - 1883). Smith, who is best known for the Smith normal form of a matrix, was a professor at Oxford who made great contributions in matrix theory and number theory. In this project, we will explore parts of a paper he wrote titled On the Integration of Discontinuous Functions.