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.
 A: I think there is a subtle difference. In the first place, I want to thank the author John Lee of Introduction to Topological Manifolds, which clarifies the two notions. All the excerpts below are from this book.
The following excerpt gave a definition of a universal property: categorical product.

Then the example (b) in the following excerpt gave an example of a product space in the product topology. It said that the product space is the categorical product in the category Top

The characteristic property of product topology is given below

Here is the point, note the words above the red line in the 2nd excerpt. When we prove the characteristic property of product topology, we use only topological methods. Then, based on this characteristic property, we get that a product space with the product topology satisfies the language of universal property. In other words, it can be interpreted as a universal property in the category theory. In short, we can use characteristic property(a property has nothing to do with category theory), to construct many things, guaranteeing these things satisfying the universal property.
