I About the definition of topological manifold using closed sets

[cianfa72]

TL;DR

How to define the notion of topological vs smooth manifold using closed sets.

We all know the definition of n-dimensional topological manifold uses open sets and homeomorphisms onto the image as open set in $\mathbb R^n$.

It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on $\mathbb R^n$ ? I'm positive for this.

Perhaps the definition of smooth manifold would be problematic, though.

 

[fresh_42]

You can always replace an open set by its closed complement. Of course, this is not really a difference.

All terms in this context are about local phenomena, continuity, and differentiability. These are inevitably connected to open neighborhoods. If you make those neighborhoods closed, then you immediately get into trouble with singletons.

 

[wrobel]

How to define the notion of topological vs smooth manifold using closed sets.

for what purpose, if I may ask?

 

[cianfa72]

for what purpose, if I may ask?

Nothing. I asked it just to better understand how topological vs differential elements enter the definition of manifold (topological vs differentiable).

 

[cianfa72]

All terms in this context are about local phenomena, continuity, and differentiability. These are inevitably connected to open neighborhoods. If you make those neighborhoods closed, then you immediately get into trouble with singletons.

Sorry, can you better explain what is the trouble with singletons set ?

 

[martinbn]

It should be possible to reformulate the definition of n-dimensional topological manifold using closed sets on the manifold's topology and on $\mathbb R^n$ ? I'm positive for this.

How?

 

[fresh_42]

Sorry, can you better explain what is the trouble with singletons set ?

A singleton is a one-point set $\{p\}.$ This is a closed set (in the usual topologies that are required to perform calculus). However, there is no way to define continuity or differentiability at $p$ without having a neighborhood of it.

The most basic concept of a manifold uses continuity. But continuity means a function is a morphism in TOP. It says that you cannot gain an open set by a morphism that wasn't already in the topology (of open sets) beforehand. That's why you need open sets. You can rephrase these by closed sets, but that is a void improvement, as it only means to switch to the complements.

You will also get problems with bounded manifolds. The boundary is closed. Again, a problem for continuity unless you restrict the entire setup to the boundary itself. Same with the boundaries of possibly closed neighborhoods. You cannot define continuity there, only one-sided. But how would you patch them if only one-sided continuity is defined?

Felix Hausdorff [12] introduced an axiomatic concept of neighborhoods in 1914 [13].

Source: https://www.physicsforums.com/insights/the-many-faces-of-topology/
where you can find the references [12] and [13].