Paracompact Spaces

Toroid, via Wikivisual

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It’s strange that I was actually researching paracompact topological spaces on Sunday and that we are seeing those types of spaces in my complex analysis class. We just started the Berenstein & Gay Complex Variables¹ book and things are pretty interesting. I actually deduced that we were heading there because of some of the concepts that we are seeing.

A paracompact space is a topological space in which every open cover admits an open locally finite refinement.

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Germs & Sheaves

Hyperbolic triangle, via Wikipedia
Hyperbolic triangle, via Wikipedia

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I spent most of the day reading up on differentiable manifolds, Riemann surfaces, germs, and sheaves. Some of the concepts are extremely interesting since they tie into category theory. This led me to differential geometry. I supposed that differential geometry had more to do with Euclidean geometry, an undergrad class that I didn’t enjoy all that much³, but it’s got a lot more to do with the geometry and structure of differentiable manifolds, which interest me¹.

Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds.

The study of calculus on differentiable manifolds is known as differential geometry.

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