194 lines
9.4 KiB
Markdown
194 lines
9.4 KiB
Markdown
# Math 4201 Exam 2 Review
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> [!NOTE]
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>
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> This is a review for definitions we covered in the classes. It may serve as a cheat sheet for the exam if you are allowed to use it.
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## Connectedness and compactness of metric spaces
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### Connectedness and separation
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#### Definition of separation
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Let $X=(X,\mathcal{T})$ be a topological space. A separation of $X$ is a pair of open sets $U,V\in \mathcal{T}$ that:
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1. $U\neq \emptyset$ and $V\neq \emptyset$ (that also equivalent to $U\neq X$ and $V\neq X$)
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2. $U\cap V=\emptyset$
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3. $X=U\cup V$ ($\forall x\in X$, $x\in U$ or $x\in V$)
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Some interesting corollary:
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- Any non-trivial (not $\emptyset$ or $X$) clopen set can create a separation.
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- Proof: Let $U$ be a non-trivial clopen set. Then $U$ and $U^c$ are disjoint open sets whose union is $X$.
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- For subspace $Y\subset X$, a separation of $Y$ is a pair of open sets $U,V\in \mathcal{T}_Y$ such that:
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1. $U\neq \emptyset$ and $V\neq \emptyset$ (that also equivalent to $U\neq Y$ and $V\neq Y$)
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2. $U\cap V=\emptyset$
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3. $Y=U\cup V$ ($\forall y\in Y$, $y\in U$ or $y\in V$)
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- If $\overline{A}$ is closure of $A$ in $X$, same for $\overline{B}$, then the closure of $A$ in $Y$ is $\overline{A}\cap Y$ and the closure of $B$ in $Y$ is $\overline{B}\cap Y$. Then for separation $U,V$ of $Y$, $\overline{A}\cap B=A\cap \overline{B}=\emptyset$.
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#### Definition of connectedness
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A topological space $X$ is connected if there is no separation of $X$.
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> [!TIP]
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>
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> Connectedness is a local property. (That is, even the big space is connected, the subspace may not be connected. Consider $\mathbb{R}$ with the usual metric. $\mathbb{R}$ is connected, but $\mathbb{R}\setminus\{0\}$ is not connected.)
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>
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> Connectedness is a topological property. (That is, if $X$ and $Y$ are homeomorphic, then $X$ is connected if and only if $Y$ is connected. Consider if not, then separation of $X$ gives a separation of $Y$.)
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#### Lemma of connected subspace
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If $A,B$ is a separation of a topological space $X$, and $Y\subseteq X$ is a **connected** subspace with subspace topology, then $Y$ is either contained in $A$ or $B$.
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_Easy to prove by contradiction. Try to construct a separation of $Y$._
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#### Theorem of connectedness of union of connected subsets
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Let $\{A_\alpha\}_{\alpha\in I}$ be a collection of connected subsets of a topological space $X$ such that $\bigcap_{\alpha\in I} A_\alpha$ is non-empty. Then $\bigcup_{\alpha\in I} A_\alpha$ is connected.
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_Easy to prove by lemma of connected subspace._
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#### Lemma of compressing connectedness
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Let $A\subseteq X$ be a connected subspace of a topological space $X$ and $A\subseteq B\subseteq \overline{A}$. Then $B$ is connected.
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_Easy to prove by lemma of connected subspace. Suppose $C,D$ is a separation of $B$, then $A$ lies completely in either $C$ or $D$. Without loss of generality, assume $A\subseteq C$. Then $\overline{A}\subseteq\overline{C}$ and $\overline{A}\cap D=\emptyset$ (from $\overline{C}\cap D=\emptyset$ by closure of $A$). (contradiction that $D$ is nonempty) So $D$ is disjoint from $\overline{A}$, and hence from $B$. Therefore, $B$ is connected._
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#### Theorem of connected product space
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Any finite cartesian product of connected spaces is connected.
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_Prove using the union of connected subsets theorem. Using fiber bundle like structure union with non-empty intersection._
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### Application of connectedness in real numbers
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Real numbers are connected.
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Using the least upper bound and greatest lower bound property, we can prove that any interval in real numbers is connected.
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#### Intermediate Value Theorem
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Let $f:[a,b]\to \mathbb{R}$ be continuous. If $c\in\mathbb{R}$ is such that $f(a)<c<f(b)$, then there exists $x\in [a,b]$ such that $f(x)=c$.
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_If false, then we can use the disjoint interval with projective map to create a separation of $[a,b]$._
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#### Definition of path-connected space
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A topological space $X$ is path-connected if for any two points $x,x'\in X$, there is a continuous map $\gamma:[0,1]\to X$ such that $\gamma(0)=x$ and $\gamma(1)=x'$. Any such continuous map is called a path from $x$ to $x'$.
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- Every connected space is path-connected.
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- The converse may not be true, consider the topologists' sine curve.
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### Compactness
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#### Definition of compactness via open cover and finite subcover
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Let $X=(X,\mathcal{T})$ be a topological space. An open cover of $X$ is $\mathcal{A}\subset \mathcal{T}$ such that $X=\bigcup_{A\in \mathcal{A}} A$. A finite subcover of $\mathcal{A}$ is a finite subset of $\mathcal{A}$ that covers $X$.
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$X$ is compact if every open cover of $X$ has a finite subcover (i.e. $X=\bigcup_{A\in \mathcal{A}} A\implies \exists \mathcal{A}'\subset \mathcal{A}$ finite such that $X=\bigcup_{A\in \mathcal{A}'} A$).
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#### Definition of compactness via finite intersection property
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A collection $\{C_\alpha\}_{\alpha\in I}$ of subsets of a set $X$ has finite intersection property if for every finite subcollection $\{C_{\alpha_1}, ..., C_{\alpha_n}\}$ of $\{C_\alpha\}_{\alpha\in I}$, we have $\bigcap_{i=1}^n C_{\alpha_i}\neq \emptyset$.
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Let $X=(X,\mathcal{T})$ be a topological space. $X$ is compact if every collection $\{Z_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ satisfies the finite intersection property has a non-empty intersection (i.e. $\forall \{Z_{\alpha_1}, ..., Z_{\alpha_n}\}\subset \{Z_\alpha\}_{\alpha\in I}, \bigcap_{i=1}^n Z_{\alpha_i} \neq \emptyset\implies \bigcap_{\alpha\in I} Z_\alpha \neq \emptyset$).
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#### Compactness is a local property
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Let $X$ be a topological space. A subset $Y\subseteq X$ is compact if and only if every open covering of $Y$ (set open in $X$) has a finite subcovering of $Y$.
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- A space $X$ is compact but the subspace may not be compact.
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- Consider $X=[0,1]$ and $Y=[0,1/2)$. $Y$ is not compact because the open cover $\{(0,1/n):n\in \mathbb{N}\}$ does not have a finite subcover.
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- A compact subspace may live in a space that is not compact.
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- Consider $X=\mathbb{R}$ and $Y=[0,1]$. $Y$ is compact but $X$ is not compact.
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#### Closed subspaces of compact spaces
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A closed subspace of a compact space is compact.
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A compact subspace of Hausdorff space is closed.
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_Each point not in the closed set have disjoint open neighborhoods with the closed set in Hausdorff space._
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#### Theorem of compact subspaces with Hausdorff property
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If $Y$ is compact subspace of a **Hausdorff space** $X$, $x_0\in X-Y$, then there are disjoint open neighborhoods $U,V\subseteq X$ such that $x_0\in U$ and $Y\subseteq V$.
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#### Image of compact space under continuous map is compact
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Let $f:X\to Y$ be a continuous map and $X$ is compact. Then $f(X)$ is compact.
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#### Tube lemma
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Let $X,Y$ be topological spaces and $Y$ is compact. Let $N\subseteq X\times Y$ be an open set contains $X\times \{y_0\}$ for $y_0\in Y$. Then there exists an open set $W\subseteq Y$ is open containing $y_0$ such that $N$ contains $X\times W$.
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_Apply the finite intersection property of open sets in $X\times Y$. Projection map is continuous._
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#### Product of compact spaces is compact
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Let $X,Y$ be compact spaces, then $X\times Y$ is compact.
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Any finite product of compact spaces is compact.
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### Compact subspaces of real numbers
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#### Every closed and bounded subset of real numbers is compact
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$[a,b]$ is compact in $\mathbb{R}$ with standard topology.
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#### Good news for real numbers
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Any of the three properties is equivalent for subsets of real numbers (product of real numbers):
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1. $A\subseteq \mathbb{R}^n$ is closed and bounded (with respect to the standard metric or spherical metric on $\mathbb{R}^n$).
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2. $A\subseteq \mathbb{R}^n$ is compact.
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#### Extreme value theorem
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If $f:X\to \mathbb{R}$ is continuous map with $X$ being compact. Then $f$ attains its minimum and maximum. (there exists $x_m,x_M\in X$ such that $f(x_m)\leq f(x)\leq f(x_M)$ for all $x\in X$)
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#### Lebesgue number lemma
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For a compact metric space $(X,d)$ and an open covering $\{U_\alpha\}_{\alpha\in I}$ of $X$. Then there is $\delta>0$ such that for every subset $A\subseteq X$ with diameter less than $\delta$, there is $\alpha\in I$ such that $A\subseteq U_\alpha$.
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_Apply the extreme value theorem over the mapping of the averaging function for distance of points to the $X-U_\alpha$. Find minimum radius of balls that have some $U_\alpha$ containing the ball._
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#### Definition for uniform continuous function
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$f$ is uniformly continuous if for any $\epsilon > 0$, there exists $\delta > 0$ such that for any $x_1,x_2\in X$, if $d(x_1,x_2)<\delta$, then $d(f(x_1),f(x_2))<\epsilon$.
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#### Theorem of uniform continuous function
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Let $f:X\to Y$ be a continuous map between two metric spaces. If $X$ is compact, then $f$ is uniformly continuous.
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#### Definition of isolated point
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A point $x\in X$ is an isolated point if $\{x\}$ is an open subset of $X$.
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#### Theorem of isolated point in compact spaces
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Let $X$ be a nonempty compact Hausdorff space. If $X$ has no isolated points, then $X$ is uncountable.
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_Proof using infinite nested closed intervals should be nonempty._
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### Variation of compactness
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#### Limit point compactness
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A topological space $X$ is limit point compact if every infinite subset of $X$ has a limit point in $X$.
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- Every compact space is limit point compact.
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#### Sequentially compact
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A topological space $X$ is sequentially compact if every sequence in $X$ has a convergent subsequence.
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- Every compact space is sequentially compact.
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#### Equivalence of three in metrizable spaces
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If $X$ is a metrizable space, then the following are equivalent:
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1. $X$ is compact.
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2. $X$ is limit point compact.
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3. $X$ is sequentially compact. |