Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed here.) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.

Your third question was asked, closed, and deleted once; I started a blog discussion about it here.

Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed here.) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.

Your third question was asked, closed, and deleted once; I started a blog discussion about it here.

Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.

Your third question was asked, closed, and deleted once; I started a blog discussion about it here.

Open sets can be identified with maps from a space to the Sierpinski space, and maps out of a space pull back under morphisms. (In other words, if you believe that the essence of what it means to be a topological space has to do with functions out of the space, you are privileging inverse images over images. A related question was discussed here.) I think essentially this kind of reasoning underlies the basic appearances of inverse images in mathematics. For example, in the category of sets, subsets can be identified with maps from a set to the two-point set, and again these maps pull back under morphisms. This should be responsible for the nice properties of inverse image with respect to Boolean operations.

Your third question was asked, closed, and deleted once; I started a blog discussion about it here.