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--------------------

Solutions:

Definitions:

1. Nhood of a Point in a Topological Space:

   Let \((X,\tau)\) be a topological space, and \(x \in X\). A neighborhood of \(X\) is a set \(U\subseteq X\) which contains \(x\) and which contains an open set containing \(x\). So \(x \in V \subseteq U \subseteq X\), and \(V \in \tau\).

   Example:

   In \(\mathbb{R}^2\), the open unit square centered at \((0,0)\) is a neighborhood of \((0,0)\).

   Non-Example:

2. Nhood System at a Point in a Topological Space:

   Let \((X,\tau)\) be a topological space, and \(x \in X\). A neighborhood system of \(x\) is the set of all neighborhoods of \(x\).

   Example:

   In \(\mathbb{R}^2\), the neighborhood system of \((0,0)\) is every set \(P\) such that \(P\) has a non-empty intersection with some open disk centered at \((0,0)\).

   Non-Example:

3. Nhood Base at a Point in a Topological Space:

   A neighborhood base at \(x \in X\) is a collection \(B\) of open sets containing \(x\), such that every open set containing \(x\) is a superset of a member of \(B\).$$

\forall S \in tau: x \in S \rightarrow \exists B_{lambda} \in B: B_{\lambda} \subseteq S