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A First Course in Complex Analysis by Matthias Beck, Gerald Marchesi, Dennis Pixton, and Lucas Sabalka Exer 1.29,30

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Definitions:

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If I understood the either, or, and, etc right, then $G=(-2,-1) \cup D[0,1] \cup \{1,2\}$ and thus

  • 1.29 (a) $G$ is from left to right, an interval in $\mathbb R$, the unit disc, a point in $\mathbb R$ and then another point in $\mathbb R$.
  • 1.29 (b) Interior: $D[0,1]$
  • 1.29 (c) Boundary: $[-2,-1] \cup C[0,1] \cup \{1,2\} = [-2,-1) \cup C[0,1] \cup \{2\}$
  • 1.29 (d) Isolated: $\{2\}$

1.30

Let $A=(-2,-1), B=D[0,1],C=\{1\},D=\{2\}$. I'm gonna group these into $G_1$ and $G_2$ separated s.t. $G_1 \subseteq G_3$ and $G_2 \subseteq G_4$

Ways:

  1. $G_1=A \cup B \cup C, G_2 = D, G_3 = \{x < 1.5\}, G_4 = \{x > 1.5\}$
  2. $G_1=A \cup D, G_2 = B \cup C, G_3 = \{x < -1\} \cup \{x > 1.5\}, G_4 = \{-1 < x < 1.5\}$
  3. $G_1=A, G_2 = B \cup C \cup D, G_3 = \{x < -1\}, G_4 = \{x > -1\}$

Where have I gone wrong please?

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  • $\begingroup$ Did they really write $-2<z<-1$? I'd get another book.... $\endgroup$ Jul 29, 2018 at 11:56
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    $\begingroup$ @LordSharktheUnknown Why? It's a perfectly valid property of $z \in \mathbb{C}$ to say that $z$ is real and $-2 < z < -1$. $\endgroup$
    – Adayah
    Jul 29, 2018 at 11:58
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    $\begingroup$ Two places. First, note a boundary point of G need not be a member of G. Look again at -2. Second, none of your separations split B and C. That's correct, but suggests you should look again at the isolation of 1. $\endgroup$ Jul 29, 2018 at 12:15
  • $\begingroup$ @DavidHartley Thanks! ^-^ I actually noticed those transcribed incorrectly. Post as answer? $\endgroup$
    – BCLC
    Jul 29, 2018 at 12:17
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    $\begingroup$ What does the notation $D[0,1]$ mean? $\endgroup$ Jul 29, 2018 at 13:40

1 Answer 1

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Two places. First, note a boundary point of G need not be a member of G. Look again at -2. Second, none of your separations split B and C. That's correct, but suggests you should look again at the isolation of 1. – David Hartley

@DavidHartley Thanks! ^-^ I actually noticed those transcribed incorrectly. Post as answer? – BCLC

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