Math 440, Fall 2014, Assignment 3

By one of those caprices of the mind, which we are perhaps most subject to in early youth, I at once gave up my former occupations; set down natural history and all its progeny as a deformed and abortive creation; and entertained the greatest disdain for a would-be science, which could never even step within the threshold of real knowledge. In this mood of mind I betook myself to the mathematics, and the branches of study appertaining to that science, as being built upon secure foundations, and so, worthy of my consideration.

- Mary Shelley, Frankenstein

Carefully define the following terms, then give one example and one non-example of each:

1. Nhood (of a point $x$ in a topological space $(X,\tau)$).
2. Nhood system (at a point $x$ in a topological space $(X,\tau)$).
3. Nhood base (at a point $x$ in a topological spae $(X,\tau)$).
4. Basic nhood (of $x$).
5. Base (for a topology).
6. Basic open set (in $X$).
7. Subbase (for a topology).

Carefully state the following theorems (you do not need to prove them):

1. Theorem characterizing nhood systems (Theorem 4.2).
2. Theorem characterizing nhood bases (Theorem 4.5).
3. Theorem characterizing bases (Theorem 5.3).
4. Theorem characterizing subbases (Theorem 5.6).

Solve the following problems:

1. Problems 4A, 4D, 4F, 5A, and 5B.

--------------------End of assignment--------------------