What is the meaning of the term “inductively P map”?

In this page is the definition of an inductively open map. But in this pdf is the definition of a inductively P map, where P is a property of maps.

But there is a difference in the definitions. In the former it's said that "the image of the subset is the same as the image of the whole space," but in the other it's said symbolically that the image of the subset is the entire codomain space.

It's more credible that the terminology of the paper is the correct, but in some papers is used nonstandard terminology and I'm a little bit confused. What is the meaning of the term "inductively P map" is the sense of the paper standard or is the other page correct? and in any case what is the terminology of the other case?

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a) Please don't use exclamation marks like that in the title; imagine what the main page would look like if everyone did that. No information is being conveyed in that punctuation. b) Please make the title more specific. "What is the correct term?" would fit very many terminology questions. In fact it doesn't even really fit yours, since you're mainly looking for the meaning of a term you already know. Why not use a title like e.g. "What's an inductively $\mathcal P$ map?" –  joriki Aug 20 '12 at 3:53