# Characteristic Property = Universal Property?

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.

• We often characterize objects by a universal property they satisfy; such characterizations will naturally look like universal properties! – user14972 Mar 22 '14 at 16:48
• Yes, I know, but if u compare carefully u will notice some discrepancies – C-Star-W-Star Mar 22 '14 at 16:58
• 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. – Stefan Hamcke Mar 22 '14 at 17:19