Let be a smooth map between two manifolds. We call a regular point provided the derivative is non singular, so that there exists a neighbourhood of such that it is diffeomorphically mapped onto its image.(See Inverse function theorem) A point is called a regular value if contains only regular points. Otherwise we call them crtitical […]
Category: Topology
Tangent space, Derivative
In the previous post we have seen some examples of manifolds and the most general ways in which they occur. By now we know what we mean by differentiable functions on the differentiable manifolds. But we have not defined what do we mean by derivative of a function. So the the natural way to do […]
Differentiable structures-II
In the previous post we have defined a differentiable structure.The definition allows us to talk of differentiable functions i.e., we gave meaning to the term ” f is differentiable function on the manifold”. So basically, differentiable structure is the topological manifold together with sets of differentiable functions where the sets satisfy some properties. For instance […]
Differentiable structures
I am taking a course on differential topology this semester and will be writing a series of posts on this topic. In this post, we will look at the notion of differentiable structures on a topological manifold. Differentiable structure on a manifold is a basic setting in which we can talk of differentiability of functions […]
Topological Spaces and Continuity
Our intuitive understanding of continuity is that, a function is continuous at when points close to are mapped to points close to When we have a notion of distance, for points which get closer and closer to x, the corresponding values of a function should get closer and closer to . Thus, we can formulate […]
A Topological Proof of the Infinitude of Primes
The following is a topological proof of the infinitude of primes due to Furstenberg. Consider a topology on generated by family of sets . It is easy to verify that this collection forms a basis for the topology. Now every non empty open set in is infinite so that no finite set is open. Also […]
A Neighbourhood Of Infinity 