ManifoldSatisfactory

A manifold is some set of points such that for each one we can consult a chart which will transport some region of that manifold containing the point into a region of euclidean space (well understood). A country is a region of the Earth's surface. A map of a country is a chart that gives you that region of the manifold (Earth) projected onto the euclidean plane. We call a collection of charts covering the entire manifold an atlas. Moving from a manifold into euclidean space is useful because doing geometry and calculus will be much easier there.

The more typical definition is that a manifold is a space that is locally like $\mathbb{R}^n$. I dislike it philosophically, but it's easier to formalize than M1-M3 above. To formalize it, you can say that a manifold is a space in which any sufficiently small neighborhood is homeomorphic to an open set in $\mathbb{R}^n$. Homeomorphic means that you can find a homeomorphism between them. A homeomorphism is, intuitively, a process of stretching and distorting something without cutting or gluing. More formally, a homeomorphism is a function that is invertible and continuous in both directions.

ManifoldPipe

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M3. Completeness: M is complete, in the sense that specifying an arbitrarily small neighborhood gives a unique definition of a point.

Manifoldvalve

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Manifoldbetting

If you work through some of the examples collected earlier, you'll see that this pretty much does the job. You can verify that the following are manifolds: the real line, a circle, the open half-plane $y>0$, the union of two disjoint planes. And that the following are not: a line glued to a plane, the rational numbers, the closed half-plane $y \ge 0$.

Those of us who were introduced to manifolds via point set topology (as in Munkres) have a gut feeling that this is what manifolds are, and that the differential structure is an overlay. Those of us who were introduced to manifolds via the differential structure (as in Spivak) have a gut feeling that that is what manifolds are.

ManifoldMarkets

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ManifoldEngineering

Both the WP article and this book have helpful lists of things that are and aren't manifolds. I would suggest having these lists handy while going through actual definitions of manifolds, because otherwise it's hard to understand why the different aspects of the definitions make sense.

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Manifoldon a car

It's also helpful to prepare yourself with an intuitive, informal idea of what we're trying to encapsulate in a formal definition. The basic idea is that we want to be able to describe a geometry stripped of (1) any notion of measurement, and (2) any notion of what is a straight line. However, we want to preserve distinctions like the distinction between a torus and a sphere.

Does this mean that in mathematics a manifold is essentially a representation of something that otherwise would difficult to "model in another way" in order to use it for some other purpose?

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First off, the definition is not totally standardized. I ran across the following comment on the talk page of the WP article, which I thought was very helpful in explaining this possible source of confusion:

A manifold is a linear space. It is closed under addition. That is, if x are in the manifold then x+y is also in the manifold. A flat map on your desk is not a linear manifold because it is finite. Don't believe everything you read on the internet.