Topology, 2ed
| readinglist | |
|---|---|
| author | Munkres |
| summary |
A thorough dive into general and algebraic topology. |
| status | reading |
Notes
Set Theory and Logic
This chapter is a review of basic mathematical concepts, beginning with set theory. One key point to keep in mind is that Munkres will occasionally disambiguate the notation $(a,b)$ as an ordered pair versus as an interval by using $a\times b$ for the former. The text also prefers $\subset$ over $\subseteq$, emphasizing proper subsets with $\subsetneq$. I will not follow that convention in these notes.
Given a function $f:A\to B$, the text defines:
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its domain as $A$
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its range as $B$
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its image set as $f(A)$
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the restriction of $f$ to $A_0\subseteq A$, denoted $f|A_0$, as $\{(a, f(a)): a\in A_0\}$
Composites, preimages, injectivity, surjectivity, and bijectivity are all defined as usual.
Definition: order type
Suppose that $A$ and $B$ are two sets with respective order relations $<_A$ and $<_B$. We say that $A$ and $B$ have the same order type if there exists a bijection $f:A\to B$ s.t. $a_1<_A a_2 \implies f(a_1)<_B f(a_2)$.
Definition: largest element, smallest element
Suppose that $A$ is a set ordered by the relation $<$. Let $A_0\subseteq A$. We say that $b$ is the largest element of $A_0$ if $b\in A_0$ and $x\leq b~\forall x\in A_0$. Similarly, we say that $a$ is the smallest element of $A_0$ if $a\in A_0$ and $a\leq x~\forall x\in A_0$.
Definition: bounded above, upper bound, supremum, bounded below, lower bound, infimum
We say that the subset $A_0$ is bounded above if there is an element $b\in A$ s.t. $x\leq b~\forall x\in A_0$; $b$ is called an upper bound for $A_0$. If the set of all upper bounds for $A_0$ has a smallest element, that element is called the supremum, denoted $\sup A_0$.
Similarly, $A_0$ is bounded below if there is an element $a\in A$ s.t. $a\leq x~\forall x\in A_0$; $a$ is a lower bound for $A_0$. If the set of all lower bounds of $A_0$ has a largest element, that element is called the infimum, denoted $\inf A_0$.
An ordered set $A$ has the least upper bound property if each nonempty subset $A_0$ of $A$ that is bounded above has a least upper bound. Analogously, it has the greatest lower bound property if each nonempty $A_0\subseteq A$ which is bounded below has a greatest lower bound. $A$ has one of these properties iff it has the other (see Exercise 3.13).