What is inheriting of topology? What is the use of studying it?

Question

if S is the subset of M(topological space), can one construct on S a topology from topological space ,say Om or M?

Answer 1

Yes, if $M, \mathcal{T})$ is a topological space, and $S \subseteq M$ is a subset, we can make $S$ into a topological space $(S,\mathcal{T}_S)$ by defining

$$\mathcal{T}_S = \{O \cap S\mid O \in \mathcal{T}\}$$

This is a natural topology, in the sense that it is the smallest one that makes the inclusion map $i_S:S \to M, i_S(s)=s$ continuous between $S$ and $(M,\mathcal{T})$.

It allows us to talk about continuity of functions on $S$ too, etc.

In many branches of mathematics substructures (subspaces in linear algebra, subgroups in group theory, subgraphs in graph theory etc. etc.) play an important role. You can ask which spaces occur as subspaces of other spaces etc, and whether properties of the large $M$ "inherit" to subspaces $S$ (some do, some don't , some under conditions on $S$ etc.).