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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 $Y$:

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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

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

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