Is it possible to obtain the local defintion from the global definition, in Topology?

Question

Is it possible to obtain the local defintion from the global definition, in Topology?

We have the following properties:

• connectedness
• path-connectedness
• being a bases
• compact
• etc..

Each of these definitions also has a local and global defintion. But is it possible to obtain the local defintion from the global definition? Or vice versa, going from the local defintion to the global defintion? (Or should I learn both the global and local defintions by heart?)

I'm having trouble seeing the pattern of connection between the local and global defintions.

Regards, Jens

Answer 1

But is it possible to obtain the local defintion from the global definition?

Sort of. One recurring pattern is the following:

$X$ is locally $P$ at $x\in X$ if for any open neighbourhood $x\in U\subseteq X$ there is an open neighbourhood $x\in V\subseteq U$ such that $V$ has $P$.

$X$ is locally $P$ if it is locally $P$ at every point.

This however fails for local compactness where we require that "$\overline{V}$ has $P$" not "$V$ has $P$" (proper open subsets are rarely compact, never in connected Hausdorff case).

Things become even more complicated in non-Hausdorff case where different definitions of local compactness are not equivalent. Indeed, there is more then one definition of local compactness.

Or should I learn both the global and local defintions by heart?

At the end of the day that's the only reliable way.