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 \(B\).$$ \forall S \in \tau: x \in S \rightarrow \exists B_{\lambda} \in B: B_{\lambda} \subseteq S $$

   Example:

   The set of all open disks centered at \((0,0)\) with radius less than 1 is a neighborhood base at \((0,0)\).

   Non-Example:

4. Basic Nhood of \(x\):

   If \(B\) is a neighborhood base of \(x\), then an element of \(B\) is a basic neighborhood of \(x\).

   Example:

   In the previous example, he open half-disk would be a basic neighborhood of \((0,0)\).

   Non-Example:

5. Base for a Topology:

   A base for a topological space \((X,\tau)\) is a collection of set \(\mathscr{B}\) such that:$$ \bigcup_{B\in\mathscr{B}} B = X $$

   If \(B_1\) and \(B_2\) are in \(\mathscr{B}\) and \(p\in B_1 \cap B_2\), then there is some \(B_3\) such that \(p \in B_3 \subseteq B_1 \cap B_2\)
   If a set is open in \(\tau\), then it must be a union of members of \(\mathscr{B}\).

   Example:

   The set of all open disks of radius less than 1 is a base for \(\mathbb{R}^2\) with the Euclidean metric topology (? Proper terminology?)

   Non-Example:

6. Basic Open Set:

   If \(\mathscr{B}\) is a base for \(\tau\), then a basic open set is a member of \(\mathscr{B}\).

   Example:

   Non-Example:

7. Subbase for a Topology:

   A subbase for \(\tau\) is a collection of sets which, when closed under finite intersections, is a base for \(\tau\).

   Example:

   Non-Example:

Theorems:

1. Theorem Characterizing Nhood Systems

   Let \(U_x\) be the neighborhood system at \(x\). Then:
   \(U \in U_x \rightarrow x \in U\)
   \(U,V \in U_x \rightarrow U \cap V \in U_x\)
   \(U \in U_x\) implies that there is a \(V\in U_x\) such that \(U\) is a neighborhood of every point of \(V\).
   If \(U \in U_x\) and \(U \subseteq V\) then \(V \in U_x\)

2. Theorem Characterizing Nhood Bases

   Let \(B_x\) be a neighborhood base of \(x\). Then:
   \(V \in B_x \rightarrow x \in V\)
   \(V_1,V_2 \in B_x\) implies there is a \(V_3 \in B_x\) such that \(V_3 \subseteq V_1 \cap V_2\).
   If \(V \in B_x\) then there is a set \(U \in B_x\) such that, for every point \(y \in U\), there is a set \(W \in B_y\) with \(W \subseteq V\).

3. Theorem Characterizing Bases

   If \(\mathscr{B}\) is any collection of subsets of \(X\) that satisfies the first two requirements for a base, then it defines a topology by taking the open sets to be unions of elements of \(\mathscr{B}\).

4. Theorem Characterizing Subbases

   If \(X\) is a set, any collection of subsets of \(X\) is a subbase for a topology.