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5 Pages

TopologicalSpaces

Course: MATH 132, Fall 2009
School: University of Iowa
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Word Count: 1580

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2400 \parindent \magnification 0pt \parskip 12pt \hsize 7.2truein \vsize 9.7 truein \hoffset -0.45truein \def\u{\vskip -7pt} \def\v{\vskip -6pt} Defn: If $X$ is an ordered set and $a \in X$, then the following are rays in $X$: \centerline{$(a, +\infty) = \{x ~|~ x &gt; a\}, ~(-\infty, a) = \{x ~|~ x &lt; a\}$,} \centerline{$[a, +\infty) = \{x ~|~ x \geq a\}, ~(-\infty, a] = \{x ~|~ x \leq a\}.$}...

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2400 \parindent \magnification 0pt \parskip 12pt \hsize 7.2truein \vsize 9.7 truein \hoffset -0.45truein \def\u{\vskip -7pt} \def\v{\vskip -6pt} 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. \vskip 6pt \vfil \hrule \vskip -6pt \vfil 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 $X \times Y$. \vskip 6pt \vfil \hrule \vskip -6pt \vfil 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$. HW p91: 3 (prove your answer). \vskip 6pt \vfil \hrule \vskip -6pt \vfil 17. Closed Sets and Limit Points Defn: The set$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...

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V C TO A A INSr Zn a aa u i g
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'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:09:38.339 20020322-2-001_150740.jpg [020] Switches002: 00:09:38.339 00:00:37.105 20020322-2-002_150818.jpg [021] Switches003: 00:10:15.445 00:00:24.792 20020322-2-003_150
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'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:00:55.330 20020223-1-001_080526.jpg [001] + No Title +002: 00:00:55.330 00:00:01.902 20020223-1-002_080527.jpg [002] Acknowledgement003: 00:00:57.232 00:01:44.339 20020
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'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:07:49.457 20020222-1-001_135003.jpg [001] + No Title +002: 00:07:49.457 00:00:02.652 20020222-1-002_135006.jpg [002] Acknowledgement003: 00:07:52.109 00:03:14.300 20020
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'Sld#: Start_Time Duration File_Name [pg#] Title 001: 00:00:00.000 00:00:46.433 ise-el536-20020208-2-001_144123.jpg [027] Mobile Telephone002: 00:00:46.433 00:03:21.152 ise-el536-20020208-2-002_144444.jpg [028] Mobile Telephone003: 00:04:07
Allan Hancock College - MECH - 2403
SINGLE STAGE RECIPROCATING AIR COMPRESSOR TEST - RESULTSReceiver Pressure (lbf/in2 gauge) Water inlet temperature (oC) Water outlet temperature (oC) Air inlet temperature (oC) Air outlet temperature (oC) Dynamometer load (lbf) Water flow rate (l/s)
Allan Hancock College - MECH - 2403
Chapter 13: Isentropic EfficiencyAll thermo calculations assume Reversible processes Real processes never reversible Need to adjust theoretical results to correspond with actual values We base &quot;adjustments&quot; on empirical data Convenient factor used:
Allan Hancock College - MECH - 2403
Fluid Mechanics at the University of Western Australia26/7/05 9:52 AMFluid Mechanics @ UWAConservation of energy: the Bernoulli equationAims Background Procedure Questions AIMSIn flow through a tube of variable section you will investigate the
Allan Hancock College - MECH - 2403
The University of Western AustraliaSchool of Mechanical EngineeringThermoFluids 209Classwork 11A partially correct definition of properties states: &quot;The state of a system is defined by a collection of observable macroscopic quantities, ie.
Allan Hancock College - MECH - 2403
The University of Western AustraliaSchool of Mechanical EngineeringThermoFluids 209Classwork 21. Describe very briefly each of the forms of energy of particular interest in thermodynamics.2.a. b.Describe one situation where the temperatu