Open Access. Powered by Scholars. Published by Universities.®
- Keyword
-
- Topology (4)
- Derived set (3)
- Limit point (3)
- Axioms (2)
- Connectedness (2)
-
- Analysis (1)
- Cantor set (1)
- Closed (1)
- Compactness (1)
- Continuity (1)
- Continuous (1)
- Distance (1)
- Elements (1)
- Euclid (1)
- Extreme value theorem (1)
- Geometry (1)
- Invariance of domain (1)
- Metric (1)
- Neighborhoods (1)
- Nowhere dense (1)
- Open set (1)
- Open sets (1)
- Parallel postulate; hyperbolic geometry (1)
- Pythagorean Theorem (1)
- Separation (1)
- Publication
Articles 1 - 10 of 10
Full-Text Articles in Education
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Connectedness- Its Evolution And Applications, Nicholas A. Scoville
Topology
No abstract provided.
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
From Sets To Metric Spaces To Topological Spaces, Nicholas A. Scoville
Topology
No abstract provided.
Nearness Without Distance, Nicholas A. Scoville
A Compact Introduction To A Generalized Extreme Value Theorem, Nicholas A. Scoville
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.
The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville
The Closure Operation As The Foundation Of Topology, Nicholas A. Scoville
Topology
No abstract provided.
The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder
The Failure Of The Euclidean Parallel Postulate And Distance In Hyperbolic Geometry, Jerry Lodder
Geometry
No abstract provided.
Connecting Connectedness, Nicholas A. Scoville
The Cantor Set Before Cantor, Nicholas A. Scoville
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.
Topology From Analysis, Nicholas A. Scoville
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.
The Exigency Of The Euclidean Parallel Postulate And The Pythagorean Theorem, Jerry Lodder
The Exigency Of The Euclidean Parallel Postulate And The Pythagorean Theorem, Jerry Lodder
Geometry
No abstract provided.