1) A syllogism is a logical argument. In two statements, call 'premises', terms are related in categorical terms. In the classic example;

All animals are mortal.
and; All humans are animals.
then; All humans are mortal.

"animal" is a categorie of "mortal". Also, "human" is a categorie of "animal". Thus, the conclusion of the syllogism is that "human" is a categorie of "mortal".

2) On page 3, Barr talks about a line being like a string, where it can be manipulated into many shapes without individual points that make up the line changing in order. Going further, the string could even concieved as an elastic band, where the distance between the points can possibly change.

Every arrangement of that string or elastic band has its own "geometry", but its "topology" is unchanging, provided the string isn't cut or attached back onto itself.

3) In many stages of design development, architecture is looked at in purely topological terms. For example, a program diagram may show the different building spaces using vague (bubble) shapes and lines that represent physical connections. This allows the designer to begin to plan the spaces without introducing specific geometry.