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--- reading:topology [2023-05-30 00:37] – asdf
+++ reading:topology [2023-06-12 04:42] (current) – [Set Theory and Logic] corrected a typo asdf
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 ===== 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 latter. The text also prefers $\subset$ over $\subseteq$, emphasizing proper subsets with $\subsetneq$. I will not follow that convention in these notes.
+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:
@@ Line 38: / Line 38: @@
 <WRAP center round box 80%>
-**Definition**
+**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. $a_1<_A a_2 \implies f(a_1)<_B f(a_2)$.
 </WRAP>
+<WRAP center round box 80%>
+**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 $a\leq x~\forall x\in A_0$.
+</WRAP>
+<WRAP center round box 80%>
+**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 $\inf A_0$.
+</WRAP>
+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).
+++++ Exercise 3.13 |
+> Suppose that an ordered set $A$ has the least upper bound property. Prove that it has the greatest lower bound property.
+++++
 {{tag>math topology algebra}}