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Problem

They seem to be the same -almost! But are they really or is it just unlucky accident that they look so similar however describe totally different notions?

Example

I was trying to set the characteristic property of the initial resp. final topology into the framework of initial resp. terminal objects; however, I wasn't able to do so. So my question arose wether they are really the same.

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    $\begingroup$ We often characterize objects by a universal property they satisfy; such characterizations will naturally look like universal properties! $\endgroup$ – user14972 Mar 22 '14 at 16:48
  • $\begingroup$ Yes, I know, but if u compare carefully u will notice some discrepancies $\endgroup$ – C-Star-W-Star Mar 22 '14 at 16:58
  • $\begingroup$ Given a map $f:X\to Y$ from a space to a set $Y$, the final topology $\tau_f$ together with the continuous $f:X\to (Y,\tau_f)$ is an initial object in a certain category. $\endgroup$ – Stefan Hamcke Mar 22 '14 at 17:19
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No they're not!!!

A universal property defines one object uniquely while a characteristic property defines many objects uniquely (it needs a pair as input).

See the discussion in Categorical Product vs Topological Product!

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