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Office hours:

  • Mondays 18:00-19:00
  • Wednesdays 14:00-16:00

Email

GSI:

Nima Moini
  • Mondays 10:00-14:00
  • Tuesdays 10:00-14:00
  • Wednesdays 10:00-12:00

Exams

  1. Suppose that $X, Y, Z$ are metric spaces, and $f: X \to Y$ and $g : Y \to Z$ are so that $g$ is continuous and $g \circ f$ is continuous. Then $f$ must be continuous.
    1. True.
    2. False.
  2. Suppose that $X, Y$ are metric spaces and $f : X \to Y$ has the property that every open set $U \subseteq X$ has open image $f(U) \subseteq Y$. Then...
    1. ...$f$ is continuous.
    2. ...$f$ is continuous, provided that $X$ is compact.
    3. ...$f$ is continuous, provided that both $X$ and $Y$ are compact.
    4. ...$f$ need not be continuous, even if $X$ and $Y$ are compact.
  3. Suppose that $X, Y$ are metric spaces and $f : X \to Y$ has the property that every closed set $F \subseteq Y$ has closed preimage $f^{-1}(F) \subseteq X$. Then...
    1. ...$f$ is continuous.
    2. ...$f$ is continuous, provided that $X$ is compact.
    3. ...$f$ is continuous, provided that both $X$ and $Y$ are compact.
    4. ...$f$ need not be continuous, even if $X$ and $Y$ are compact.
  4. Suppose that $X, Y$ are metric spaces and $f : X \to Y$ is continuous. The $f$ is uniformly continuous, provided...
    1. ...$X$ and $Y$ are compact.
    2. ...$X$ is compact.
    3. ...$Y$ is compact.
    4. ...at least one of $X$ and $Y$ is compact.

  5. We want to prove that if a subset $E$ of a metric space $X$ is connected, then so is $\overline E$. Is the following proof valid?

    Suppose that $U_1, U_2$ are a disconnection of $\overline E$. Then since $\overline E \cap U_1 \neq \emptyset$, it must be the case that $E \cap U_1$ is non-empty (for otherwise $U_1^c$ is a closed set containing $E$, and so $\overline{E} \subseteq U_1^c$). Likewise, $E \cap U_2$ is non-empty. Since $E \subseteq \overline E \subseteq U_1\cup U_2$, and by assumption $U_1 \cap U_2 = \emptyset$, these give a disconnection of $E$.

    1. Yes
    2. No