2400
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Defn: If $X$ is an ordered set and $a \in X$, then the following are rays in $X$:
\centerline{$(a, +\infty) = \{x ~|~ x > a\}, ~(-\infty, a) = \{x ~|~ x < a\}$,}
\centerline{$[a, +\infty) = \{x ~|~ x \geq a\}, ~(-\infty, a] = \{x ~|~ x \leq a\}.$}
Lemma: The collection of all open rays is a subbasis for the order topology.
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15: The Product Topology
Let ${\cal T}_X$ denote the topology on $X$ and ${\cal T}_Y$ denote the topology on $Y$.
Defn: Let $X$ and $Y$ be topological Spaces. The {\bf product topology} on $X \times Y$ is the topology having as basis ${\cal B } = \{U \times V ~|~ U \in {\cal T}_X , V \in {\cal T}_Y \}$.
Thm 15.1: If ${\cal B}_X$ is a basis for the topology of $X$ and ${\cal B}_Y$ is a basis for the topology of $Y$, then ${\cal D} = \{ U \times V ~|~ U \in {\cal B}_X , V \in {\cal B}_Y \}$ is a basis for the topology of $X \times Y$.
Ex. 1: If $R$ has the standard topology, the product topology on $R \times R$ is the standard topology on $R^2$.
Defn: Let $\pi_1: X_1 \times X_2 \rightarrow X_1$, $\pi_1(x_1, x_2) = x_1$.
\break
$\pi_1$ is the projection of $X_1 \times X_2$ onto the first
component.
Note: If $U \subset X_1$, then $\pi_{1}^{-1}(U) = U \times X_2$. Thus
if
$U$ is open in $X_1$, then $\pi_{1}^{-1}(U)$ is open in $ X_1 \times
X_2$
Note: $\pi_{1}^{-1}(U) \cap \pi_{2}^{-1}(V) = U \times V$
Thm 15.2: The collection \hskip -20pt $${\cal S} = \{\pi_{1}^{-1}(U)
~|~ U
\hbox{ open in}
X
\} \cup \{\pi_{2}^{-1}(V) ~|~ V \hbox{ open in} Y \}$$ is a subbasis
for the
product topology on $X \times Y$.
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16. The Subspace Topology.
Defn: Let $(X, {\cal T})$ be a topological space, $Y \subset X$.
Then the {\bf subspace topology} on $Y$ is the set $${\cal T}_Y = \{U
\cap Y ~|~ U \in {\cal T} \}$$ $(Y, {\cal T}_Y)$ is a {\bf subspace} of
$X$.
Lemma 16.1: If ${\cal B}$ is a basis for the topology of $X$, then the
set $${\cal B}_Y = \{B \cap Y ~|~ B \in {\cal B} \}$$ is a basis for
the subspace topology on $Y$.
Lemma 16.2: Let $Y$ be a subspace of $X$. If $U$ is open in $Y$ and
$Y$ is open in $X$, then $U$ is open in $X$.
Lemma 16.3: If $A_j$ is a subspace of $X_j$, $j = 1, 2$, then the
product topology on $A_1 \times A_2$ is the same as the topology $A_1
\times A_2$ inherits as a subspace of $X_1 \times X_2$.
Note: Suppose $Y \subset X$ where $X$ is an ordered set with the order
topology. The order topology on $Y$ need not be the same as the
subspace topology on $Y$
Ex 1:$(0, 1) \cup \{5\}
Defn: Suppose $Y \subset X$ where $X$ is an ordered set. $Y$ is {\bf convex} if for all $a, b \in Y$ such that $a < b$, then $(a, b) \subset Y$
Ex. 1: $(1, 2) \cup (3, 4) \subset R$.
Ex. 2: $(1, 2) \cup (3, 4) \subset (1, 2) \cup (3, 9)$.
Lemma 16.4: Let $X$ is an ordered set with the order topology. Let $Y$ be a convex subset of $X$. Then the order topology on $Y$ is the same as the subspace topology on $Y$.
HW p91: 3 (prove your answer).
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17. Closed Sets and Limit Points
Defn: The set $A$ is {\bf closed} iff $X - A$ is open.
Thm 17.1: $X$ be a topological space if and only if the following conditions hold:
\v
(1) $\emptyset$, $X$ are closed.
\v
(2) Arbitrary intersections of closed sets are closed.
\v
(3) Finite unions of closed sets are closed.
Thm 17.2: Let $Y$ be a subspace of $X$. Then a set $A$ is closed in $Y$ if and only if it equals the intersection of a closed set of $X$ with $Y$.
Thm 17.3: Let $Y$ be a subspace of $X$. If $A$ is closed in $Y$ and $Y$ is closed in $X$, then $A$ is closed in $X$.
Defn: The {\bf interior} of $A$ = $Int~A$ = $A^0$ = $\cup_{U^open \subset A}U$
Defn: The {\bf closure} of $A$ = $Cl~A = \overline{A} = \cup_{A \subset F^{closed}} F$
Thm 17.4: Let $Y$ be a subspace of $X$, $A \subset Y$. Let $\overline{A}$ denote the closure of $A$ in $X$. Then the closure of $A$ in $Y$ equals $\overline{A} \cap Y$.
Defn: $A$ {\bf intersects} $B$ is $A \cap B \not= emptyset$
Thm 17.5: Let $A$ be a subset of the topological space $X$.
(a) $x \in \overline{A}$ and if only if ($x \in U^{open}$ implies $U
\cap A \not=\emptyset$).
(b) $x \in \overline{A}$ if and only if ($x \in B$ where $B$ is a basis element implies $B \cap A \not=\emptyset$).
neighborhood
Ex
Defn: $x \in X$ is a {\bf limit point} of $A$ iff $U \cap A \ \{x\} \not=emptyset$ for every open set $U$.
Defn: $A'$ = the set of all limit points of $A$.
Thm 17.6: $\overline{A} = A \cup A'$.
Cor 17.7: $A$ closed if and only if $A' \subset A$.
Defn: $x_n$ converges to a limit $x$ if for every neighborhood $U$ of $x$, there exists a positive integer $N$ such that $n \geq N$ implies $x_n \in U$.
Note: limit point of a set is not the same as limit of a sequence.
Defn: $X$ is {\bf Hausdorff space} if for all $x_1, x_2 \in X$ such that $x_1 \not= x_2$, there exists neighborhoods $U_1$ and $U_2$ of $x_1$ and $x_2$, respectively such that $U_1 \cap U_2 = \emptyset$.
Thm 17.8: Every finite point set in a Hausdorff space $X$ is closed.
Defn: $X$ is $T_1$ if every one point set is closed.
Thm 17.9: Let $X$ by $T_1$, $A \subset X$. Then $x$ is a limit point of $A$ is and only if every neighborhood of $x$ contains infinitely many points of $A$.
Thm 17.10: If $X$ is Hausdorff, then a sequence of points of $X$ convereges to at most one point of $X$. (IFF?)
${1 \over n}$ converges to in the finite complement topology.
Thm 17.11: If $X$ has the order topology, then $X$ is Hausdorff. The product of two Hausdorff spaces is Hausdorff. A subspace of a Hausdorff space is Hausdorff.
17. Continuous Functions
Defn: $f^{-1}(V) = \{x ~|~ f(x) \in V \}$.
Defn: $f: X \rightarrow Y$ is continuous iff for every $V$ open in $Y$, $f^{-1}(V)$ is open in $X$.
Lemma: $f$ continuous if and only if for every basis element $B$, $f^{-1}(B)$ is open in $X$.
Lemma: $f$ continuous if and only if for every subbasis element $S$, $f^{-1}(S)$ is open in $X$.
Thm 18.1: Let $f: X \rightarrow Y$. Then the following are equivalent:
(1) $f$ is continuous.
(2) For every subset $A$ of $X$, $f(\overline{A}) \subset \overline{f(A)}$.
(3) For every closed set $B$ of $Y$, $f^{-1}(B) is closed in X.
(4) For each $x \in X$ and each neighborhood $V$ of $f(x)$, there is a neighborhood $U$ of $x$ such that $f(U) \subset V$.
Defn: $f: X \rightarrow Y$ is a homeomorphism iff $f$ is a bijection and both $f$ and $f^{-1}$ is continuous.
topological property
imbedding
Thm 18.2
(a.) (Constant function) The constant map $f: X \rightarrow Y$, $f(x) = y_0$ is continuous.
(b.) (Inclusion) If $A$ is a subspace of $X$, then the inclusion ma...