# What are the application of universal property of subspace topology?

I come across following theorem:

Universal property of subspace topology: $$X$$ is any topological space $$Y\subset X$$ $$Z$$ is any another topological space if there is continuous map $$g:Z\to X$$ such that $$\operatorname{im}(g)\subset Y=\operatorname{im}(i)$$ where $$i:Y\to X$$ inclusion map. then there exist continuous map such that $$f:Z\to Y$$ such that following diagram commutes and g realises $$Z$$ as subspace of $$X$$ iff f realises $$Z$$ as subspace of $$Y$$:

I do not understand why such theorem is required?

If some one gives me motivation about such theorem named as universal theorem It would be very useful.

Any help will be appreciated

• Maybe this helps? – Henno Brandsma Jul 12 at 22:14
• This answer gives a more general setting of initial topologies, which all have universal properties. – Henno Brandsma Jul 12 at 22:17

Such a theorem implies that if $$f: X \to Y$$ is continuous and $$f[X] \subseteq Z \subseteq Y$$ and $$Z$$ gets the subspace topology wrt $$Y$$, then $$f: X \to Z$$ (so changing only the codomain) is also continuous.