|
|
| Line 1: |
Line 1: |
| − | == An Open Set == | + | ==Types of maps between metric spaces== |
| − | ===Definition===
| + | Suppose <math>(M_1,d_1)</math> and <math>(M_2,d_2)</math> are two metric spaces. |
| − | A set is said to be '''open''' in a metric space if it equals its interior (<math> A = Int(A)</math>). When we encounter topological spaces, we will generalize this definition of open.
| |
| − | However, this definition of open in metric spaces is the same as that as if we regard our
| |
| − | metric space as a topological space.
| |
| | | | |
| − | '''Properties:'''
| + | ===Continuous maps=== |
| − | # The empty-set is an open set (by definition: <math>int(\emptyset)=\emptyset</math>).
| |
| − | # An open ball is an open set.
| |
| − | # For any set B, int(B) is an open set. This is easy to see because: int(int(B))=int(B).
| |
| − | # If A,B are open, then <math>A\cap B</math> is open. Hence finite intersections of open sets are open.
| |
| − | # If <math>{A_i: i \in I}</math> (for any set if indexes I) are open, then their union <math>\cup_{i\in I} A_i</math> is open.
| |
| | | | |
| − | <U>Proof of 2:</U><BR/> | + | The map <math>f\,\colon M_1\to M_2</math> is continuous |
| − | Let <math>B_r(x)</math> be an open ball. Let <math>y \in B_r(x)</math>. Then <math>y \in B_{r-d(x,y)}(y) \subseteq B_r(x)</math>.<BR/>
| + | if it has one (and therefore all) of the following equivalent properties: |
| − | In the following drawing, the green line is <math>d(x,y)</math> and the brown line is <math>r-d(x,y)</math>. We have found a ball to contain <math>y</math> inside <math>B_r(x)</math>.
| + | ;General topological continuity: for every open set <math>U</math> in <math>M_2</math>, the preimage <math>f^{-1}[U]</math> is open in <math>M_1</math> |
| − | [[Image:Openball_is_open.svg]]
| + | :This is the general definition of continuity in topology. |
| | + | ;Sequential continuity: if <math>(x_n)</math> is a sequence in <math>M_1</math> that converges to <math>x</math>, then the sequence <math>(f(x_n))</math> converges to <math>f(x)</math> in <math>M_2</math>. |
| | + | :This is sequential continuity, due to Eduard Heine. |
| | + | ;ε-δ definition: for every <math>x\in M_1</math> and every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that for all <math>y</math> in <math>M_1</math> we have |
| | + | ::<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon.</math> |
| | + | :This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy. |
| | | | |
| − | <U>Proof of 4:</U><BR/>
| + | Moreover, <math>f</math> is continuous if and only if it is continuous on every compact subset of <math>M_1</math>. |
| − | A, B are open. we need to prove that <math>int(A \cap B) = A\cap B</math>. Because of the first propriety of int, we only need to show that <math>int(A \cap B) \supseteq A\cap B</math>, which means <math>\forall x\in A\cap B:x\in int(A \cap B)</math>. Let <math>x \in A \cap B</math>. We know also, that <math>x \in int(A), x \in int(B)</math> from the premises A, B are open and <math>x \in A, x \in B</math> . That means that there
| |
| − | are balls: <math>B_{{\epsilon}_1}(x) \subset A, B_{{\epsilon}_2}(x) \subset B</math>. Let <math>\epsilon = \min\{{\epsilon_1, \epsilon_2}\}</math>, we have that <math>B_{\epsilon}(x) \subset A, B_{\epsilon}(x) \subset B \Rightarrow B_{\epsilon}(x) \subset A\cap B </math>.
| |
| − | By the definition of an internal point we have that <math>x\in int(A \cap B)</math> (<math>B_{\epsilon}(x)</math> is the required ball).
| |
| | | | |
| − | Interestingly, this property does not hold necessarily for an infinite intersection of open sets. To see an example on the real line, let <math>A_n=\{(-1/n,1/n)\}</math>. We then see that <math>\cap^\infty_{i=1}A_i=\{0\}</math> which is closed.
| + | The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected. |
| | | | |
| − | <U>Proof of 5:</U><BR/>
| + | ===Uniformly continuous maps=== |
| − | Proving that the union of open sets is open, is rather trivial: let <math>{A_i: i \in I}</math> (for any set if indexes I) be a set of open sets.
| + | The map <math>f\,\colon M_1\to M_2</math> is uniformly continuous if for every <math>\varepsilon>0</math> there exists <math>\delta>0</math> such that |
| − | we need to prove that <math>int(\cup_{i\in I} A_i) \supseteq \cup_{i\in I} A_i</math>: If <math>x\in A_i</math> then it has a ball <math>B_\epsilon(x) \subset A_i \subseteq \cup_{i\in I} A_i</math>. The same ball that made a point an internal point in <math>A_i</math> will make it internal in
| |
| − | <math>\cup_{i\in I} A_i</math>.
| |
| | | | |
| − | '''Proposition''': A set is open, if and only if it is a union of open-balls.<BR>
| + | :<math>d_1(x,y)<\delta \implies d_2(f(x),f(y))< \varepsilon \quad\mbox{for all}\quad x,y\in M_1.</math> |
| − | '''Proof:''' Let A be an open set. by definition, if <math>x\in A</math> there there a ball <math>B_{\epsilon_x}(x) \subseteq A</math>. We can then compose A: <math>A = \cup_{x\in A}B_{\epsilon_x}(x)</math>. The equality is true because: <math> \cup_{x\in A}B_{\epsilon_x}(x) \subseteq A</math> because <math> \forall x \in A: B_{\epsilon_x}(x) \subseteq A</math>. <math> \cup_{x\in A}B_{\epsilon_x}(x) \supseteq A</math> in each ball we have the element <math>x</math> and we unite balls of all the elements of <math>A</math>. <BR>
| |
| − | On the other hand, a union of open balls is an open set, because ''every'' union of open sets is open.
| |
| | | | |
| − | ===Examples===
| + | Every uniformly continuous map <math>f\,\colon M_1\to M_2</math> is continuous. The converse is true if <math>M_1</math> is compact (Heine–Cantor theorem). |
| − | * As we have seen, every open ball is an open set.
| |
| − | * For every space <math>X</math> with the discrete metric, every set is open.
| |
| − | Proof: Let <math>U</math> be a set. we need to show, that if <math>x\in U</math> then <math>x</math> is an internal point. Lets use the ball around <math>x</math> with radius <math>\frac{1}{2}</math>. We have <math>B_\frac{1}{2}(x) = \{y\mid d(x,y) < \frac{1}{2}\} = \{x\} \subseteq U</math>. Therefore <math>x</math> is an internal point.
| |
| − | * The space <math>\mathbb{R}</math> with the regular metric. Every open segment <math>(a,b)</math> is an open set. The proof of that is similar to the proof that <math>int([a,b]) = (a,b)</math>, that we have already seen.
| |
| | | | |
| − | ===Theorem===
| + | Uniformly continuous maps turn Cauchy sequences in <math>M_1</math> into Cauchy sequences in <math>M_2</math>. For continuous maps this is generally wrong; for example, a continuous map |
| − | In any metric space <i>X</i>, the following three statements hold:<br>
| + | from the open interval <math>(0,1)</math> ''onto'' the real line turns some Cauchy sequences into unbounded sequences. |
| − | :1) The union of any number of open sets is open.<br>
| |
| − | ::<b>Proof:</b> Let <math>C</math> be a collection of open sets, and let
| |
| − | :::<math>x \in \cup C</math>. Then there exists a <math>U \in C</math> such that <math>x \in U</math>.
| |
| − | :::So there exists an <math>\epsilon>0</math> such that <math>B_\epsilon(x) \subseteq U</math>. Therefore
| |
| − | :::<math>B_\epsilon(x) \subseteq \cup C</math>.
| |
| − | :2) The intersection of a finite number of open sets is open.<br>
| |
| − | ::<b>Proof:</b> Let <math>x \in \cap C</math>, where <math>C</math> is a finite collection of open sets.
| |
| − | :::So <math>x \in U</math> for each <math>U \in C</math>. Let <math>C = {U_1,U_2,...,U_n}</math>. For each <math>i=1,2,3,...,n</math>, there exists an <math>\epsilon_i > 0</math> such that <math>B_\epsilon(x) \subseteq U_i</math>. Let <math>\epsilon = min_i</math>{<math>\epsilon_i</math>}. Therefore <math> \epsilon>0</math> and <math>B_\epsilon(x) \subseteq \cap C</math>.
| |
| − | :3) The empty set and <i>X</i> are both open.<br>
| |
| | | | |
| − | ===Theorem=== | + | ===Lipschitz-continuous maps and contractions=== |
| − | In any metric space <i>X</i>, the following statements hold:<br>
| |
| − | :1) The intersection of any number of closed sets is closed.<br>
| |
| − | :2) The union of a finite number of closed sets is closed.<br>
| |
| | | | |
| − | == Convergence ==
| + | Given a real number <math>K>0</math>, the map <math>f\,\colon M_1\to M_2</math> is ''K''-Lipschitz continuous if |
| − | === Definition ===
| |
| − | First, Lets translate the calculus definition of convergence, to the "language" of metric spaces:
| |
| − | We say that a sequence <math>x_n</math> converges to <math>x</math> if for every <math>\epsilon > 0</math> exists <math>N</math> that for each <math> n^* > N</math> the following holds: <math>d(x_{n^*},x) < \epsilon</math>. <BR/>
| |
| − | Equivalently, we can define converges using Open-balls: A sequence <math>x_n</math> converges to <math>x</math> If for every <math>\epsilon > 0</math> exists <math>N</math> that for each <math> n^* > N</math> the following holds: <math>x_{n^*} \in B_\epsilon(x)</math>.
| |
| | | | |
| − | The latter definition uses the "language" of open-balls, But we can do better - We can remove the <math>\epsilon</math> from the definition of convergence, thus making the definition more topological. Let's ''define'' that <math>x_n</math> '''converges''' to <math>x</math> (and mark <math>x_n \rightarrow x</math>) , if for '''every ball''' <math>B</math> around <math>x</math> , exists <math>N_B</math> that for each <math> n^* > N_B</math> the following holds: <math>x_{n^*} \in B(x)</math>. <math>x</math> is called the ''limit'' of the sequence.
| + | :<math>d_2(f(x),f(y))\leq K d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1.</math> |
| | | | |
| − | The definitions are all the same, but the latter uses topological terms, and can be easily converted to a topological definition later.
| + | Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general. |
| | | | |
| − | === Properties ===
| + | If <math>K<1</math>, then <math>f</math> is called a contraction. Suppose <math>M_2=M_1</math> and <math>M_1</math> is complete. If <math>f</math> is a contraction, then <math>f</math> admits a unique fixed point (Banach fixed-point theorem). If <math>M_1</math> is compact, the condition can be weakened a bit: <math>f</math> admits a unique fixed point if |
| − | * If a sequence has a limit, it has only one limit.<br/>''Proof'' Let a sequence <math>x_n</math> have two limits, <math>x\,</math> and <math>x^\prime</math>. If they are not the same, we must have <math>0<d(x,x^\prime)</math>. Let <math>\epsilon</math> be smaller than this distance. Now for some <math>N</math>, for all <math>n>N</math>, it must be the case that both <math>x_n \in B_{\epsilon / 2}(x)</math> and <math>x_n \in B_{\epsilon / 2}(x^\prime)</math> by virtue of the fact <math>x\,</math> and <math>x^\prime</math> are limits. But this is impossible; the two balls are separate. Therefore the limits are coincident, that is, the sequence has only one limit.
| |
| − | * If <math>x_n \rightarrow x</math>, then almost by definition we get that <math>d(x_n, x) \rightarrow 0</math>. (<math>d(x_n, x)</math> Is the sequence of distances).
| |
| | | | |
| − | ===Examples===
| + | :<math> d(f(x), f(y)) < d(x, y) \quad \mbox{for all} \quad x \ne y \in M_1</math>. |
| − | * In <math>\mathbb{R}</math> with the natural metric, The series <math>x_n = \frac{1}{n}</math> converges to <math>0</math>. And we note it as follows: <math>\frac{1}{n}\rightarrow 0</math>
| |
| − | * Any space, with the discrete metric. A series <math>x_n</math> converges, only if it is eventually constant. In other words: <math>x_n\rightarrow x</math> If and only if, We can find <math>N</math> that for each <math> n^* > N</math>, <math>x_{n^*} = x</math>
| |
| − | * An example you might already know:<BR/>
| |
| − | The space <math>\mathbb{R}^k</math> For any p-norm induced metric, when <math>p\geq 1</math>. Let <math>\vec{x_n} = (x_{n,1},x_{n,2},\cdots, x_{n,k})</math>. and let <math>\vec{x} = (x_{1},x_{2},\cdots, x_{k})</math>. <BR/> Then, <math>\vec{x_n} \rightarrow \vec{x}</math> If and only if <math> \forall i, 1\leq i \leq k: x_{n,i} \rightarrow x_{i}</math>.
| |
| | | | |
| − | ===Uniform Convergence=== | + | ===Isometries=== |
| − | A sequence of functions <math>\{ f_n \}</math> is said to be uniformly convergent on a set <math>S</math> if for any <math>\epsilon>0</math>, there exists an <math>N</math> such that when <math>a</math> and <math>b</math> are both greater than <math>N</math>, then <math>d(f_a(x),f_b(x)) < \epsilon</math> for any <math>x \in S</math>.
| + | The map <math>f\,\colon M_1\to M_2</math> is an isometry if |
| | + | :<math>d_2(f(x),f(y))=d_1(x,y)\quad\mbox{for all}\quad x,y\in M_1</math> |
| | + | Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open). |
| | | | |
| − | == Closed Sets == | + | ===Quasi-isometries=== |
| − | === Closure ===
| + | The map <math>f\,\colon M_1\to M_2</math> is a [[quasi-isometry]] if there exist constants <math>A\geq1</math> and <math>B\geq0</math> such that |
| − | '''Definition''': The point <math>p</math> is called a <U>point of closure</U> of a set <math>A</math> if there exists a sequence <math>a_n, \forall n, a_n \in A</math>, such that <math>a_n \rightarrow p</math>.<BR/>
| |
| | | | |
| − | In other words, the point <math>p</math> is a point of closure of a set <math>A</math> if there exists a sequence in <math>A</math> that converges on <math>p</math>. Note that <math>p</math> is not necessarily an element of the set <math>A</math>.
| + | :<math>\frac{1}{A} d_2(f(x),f(y))-B\leq d_1(x,y)\leq A d_2(f(x),f(y))+B \quad\text{ for all }\quad x,y\in M_1</math> |
| | | | |
| − | An equivalent '''definition''' using balls: The point <math>p</math> is called a <U>point of closure</U> of a set <math>A</math> if for every open ball <math>B</math> containing <math>p</math>, we have <math>B \cap A \neq \emptyset</math>. In other words, every open ball containing <math>p</math> contains at least one point in <math>A</math> that is distinct from <math>p</math>. <BR/>
| + | and a constant <math>C\geq0</math> such that every point in <math>M_2</math> has a distance at most <math>C</math> from some point in the image <math>f(M_1)</math>. |
| − | The proof is left as an exercise.
| |
| | | | |
| − | Intuitively, a point of closure is arbitrarily "close" to the set <math>A</math>. It is so close, that we can find a sequence in the set that converges to any point of closure of the set.
| + | Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in geometric group theory in relation to the word metric. |
| − | | |
| − | '''Example''': Let A be the segment <math>[0,1) \in \mathbb{R}</math>, The point <math>p = 1</math> is not in <math>A</math>, but it is a point of closure: Let <math>a_n = 1 - \frac{1}{n}</math>. <math>a_n \in A</math> (<math>n > 0</math>, and therefore <math>a_n = 1 - \frac{1}{n} < 1</math>) and <math> a_n \rightarrow 1 </math> (that's because <math> \frac{1}{n} \rightarrow 0</math>).
| |
| − | | |
| − | '''Definition''': The <U>closure</U> of a set <math>A \subseteq X</math> <math>({X},d)</math>, is the set of all points of closure. The closure of a set A is marked <math>\bar{A}</math> or <math>Cl(A)</math>.
| |
| − | | |
| − | Note that <math>A \subseteq \bar{A}</math>. a quick proof: For every <math>x \in A</math>, Let <math>(a_n = x)\forall n</math>.
| |
| − | ==== Examples ====
| |
| − | For the metric space <math>\mathbb{R}</math> (the line), and let <math>a,b \in \mathbb{R}</math> we have:
| |
| − | * <math>Cl([a,b]) = [a,b]</math>
| |
| − | * <math>Cl((a,b]) = [a,b]</math>
| |
| − | * <math>Cl([a,b)) = [a,b]</math>
| |
| − | * <math>Cl((a,b)) = [a,b]</math>
| |
| − | | |
| − | === Closed set ===
| |
| − | '''Definition''': A set <math>A \subseteq X</math> is <u>closed</u> in <math>{X}\,</math> if <math>A = Cl(A)</math>.<BR/>
| |
| − | Meaning: A set is closed, if it contains all its point of closure.
| |
| − | | |
| − | | |
| − | An equivalent '''definition''' is: A set <math>A \subseteq X</math> is <u>closed</u> in <math>{X}\,</math> If for every point <math>p \in A</math>, and for every Ball <math>B, p \in B</math>, then <math>B \cap A \neq \emptyset</math>. <BR/>
| |
| − | The proof of this definition comes directly from the former definition and the definition of convergence.
| |
| − | | |
| − | === Properties ===
| |
| − | Some basic properties of Cl (For any sets <math>A,B</math>):
| |
| − | | |
| − | * <math>A \subseteq Cl(A)</math>
| |
| − | *<math>Cl(Cl(A))=Cl(A)</math>
| |
| − | * <math>Cl(A \cup B) = Cl(A)\cup Cl(B)</math>
| |
| − | * <math>A </math> is closed iff <math>A = Cl(A)</math>
| |
| − | * While the above implies that the union of finitely many closed sets is also a closed set, the same does not necessarily hold true for the union of infinitely many closed sets. To see an example on the real line, let <math>A_n=\{[-1+\frac{1}{n},1-\frac{1}{n}]\}</math>. We see that <math>\cup_{i=1}^\infty A_i=(-1,1)</math> fails to contain its points of closure, <math>\pm 1.</math>
| |
| − | This union can therefore not be a closed subset of the real numbers.
| |
| − | | |
| − | The proofs are left to the reader as exercises. Hint for number 5: recall that
| |
| − | <math>Cl(A)=\cap \{A\subseteq S|S\text{ is closed }\!\!\}\!\!\text{ }</math>.
| |
| − | | |
| − | === Open vs Closed ===
| |
| − | That is, an open set approaches its boundary but does not include it; whereas a closed set includes every point it approaches. These two properties may seem mutually exclusive, but they are not:
| |
| − | | |
| − | * In any metric space <math>(X, d)</math>, the set <math>X</math> is both open and closed.
| |
| − | | |
| − | * In any space with a discrete metric, every set is both open and closed.
| |
| − | | |
| − | * In <math>\R</math>, under the regular metric, the only sets that are both open and closed are <math>\R</math> and <math>\emptyset</math>. However, some sets are neither open nor closed. For example, a half-open range like <math>[0, 1)</math> is neither open nor closed. As another example, the set of rationals is not open because an open ball around a rational number contains irrationals; and it is not closed because there are sequences of rational numbers that converge to irrational numbers (such as the various infinite series that converge to <math>\pi</math>).
| |
| − | | |
| − | ==== Complementary set ====
| |
| − | A Reminder/Definition: Let <math>A</math> be a set in the space <math>X</math>. We define the complement of <math>A</math>, <math>A^c</math> to be <math>X \setminus A</math>.
| |
| − | | |
| − | A Quick example: let <math>X = [0,1]; A = [0,\frac{1}{2}]</math>. Then <math>A^c = (\frac{1}{2},1]</math>.
| |
| − | | |
| − | ==== The plot continues... ====
| |
| − | A very important '''Proposition''': Let <math>A</math> be a set in the space <math>(X,d)</math>. Then, A is open iff <math>A^c</math> is closed.<BR/>
| |
| − | '''Proof:''' (<math>\Rightarrow</math>) For the first part, we assume that A is an open set. We shall show that <math> A^c = Cl(A^c)</math>. It is enough to show that <math>Cl(A^c) \subseteq A^c</math> because of the properties of closure. Let <math>p \in Cl(A^c)</math> (we will show that <math>p \in A^c</math>). <BR/>
| |
| − | for every ball <math>B, p\in B</math> we have, by definition that (*)<math>B \cap A^c \neq \emptyset</math>. If the point is not in <math>A^c</math> then <math>p \in A</math>. <math>A</math> is open and therefore, there is a ball <math>B</math>, such that: <math>p \in B \subseteq A</math>, that means that <math>B \cap A^c = \emptyset</math>, contradicting (*). <BR/>
| |
| − | (<math>\Leftarrow</math>) On the other hand, Lets a assume that <math>A^c</math> is closed, and show that <math>A</math> is open. Let <math>p</math> be a point in <math>A</math> (we will show that <math>p \in int(A)</math>). If <math>p</math> is not in <math>int(A)</math> then for every ball <math>B, p \in B</math> we have that <math>B \nsubseteq A</math>. That means that <math>B \cap A^c \neq \emptyset</math>. And by definition of closure point <math>p</math> is a closure point of <math>A^c</math> so we can say that <math>p \in Cl(A^c)</math>. <math>A^c</math> is closed, and therefore <math>p \in A^c = Cl(A^c)</math> That contradicts the assumption that <math>p \in A</math>
| |
| − | | |
| − | Note that, as mentioned earlier, a set can still be '''both''' open and closed!
| |
| − | | |
| − | === On <math>\mathbb{R}</math> ===
| |
| − | The following is an important theorem characterizing open and closed sets on <math>\mathbb{R}</math>.<br />
| |
| − | '''Theorem''': An open set <math>O</math> in <math>\mathbb{R}</math> is the union of countably many disjoint open intervals.<br />
| |
| − | Proof: Let <math>x\in O</math>. Let <math>a=\sup\{t|t\notin O, t<x\}</math> and let <math>b=\inf\{t|t\notin O, t>x\}</math>. There exists an open ball <math>(x - \epsilon, x + \epsilon)</math> such that <math>(x - \epsilon, x + \epsilon) \subseteq O</math> because <math>O</math> is open. Thus, a≤x-ε and b≥x+ε. Thus, x ∈(a,b). The set O contains all elements of (a,b) since if a number is greater than a, and less than x but is not within O, then a would not be the supremum of {t|t∉O, t<x}. Similarly, if there is a number is less than b and greater than x, but is not within O, then b would not be the infimum of {t|t∉O, t>x}. Thus, O also contains (a,x) and (x,b) and so O contains (a,b). If y≠x and y∈(a,b), then the interval constructed from this element as above would be the same. If y<a, then inf{t|t∉O, t>y} would also be less than a because there is a number between y and a which is not within O. Similarly if y>b, then sup{t|t∉O, t<y} would also be greater than b because there is a number between y and b which is not within O. Thus, all possible open intervals constructed from the above process are disjoint. The union of all such open intervals constructed from an element x is thus O, and so O is a union of disjoint open intervals. Because the rational numbers is dense in R, there is a rational number within each open interval, and since the rational numbers is countable, the open intervals themselves are also countable.
| |
| − | | |
| − | === Examples of closed sets ===
| |
| − | #In any metric space, a singleton <math>\{x\}</math> is closed. To see why, consider the open set, <math>\{x\}^c</math>. Let <math>y \in \{x\}^c</math>. Then <math>y \neq x</math>, so <math>d(y,x) > 0</math>. Let <math>\epsilon = \frac{1}{2}d(y,x)</math>. Then <math>B_\epsilon(y) \subseteq \{x\}^c</math>. So <math>\{x\}^c</math> is open, and hence <math>\{x\}</math> is closed.
| |
| − | #In any metric space, every finite set <math>T = \{x_1,x_2,...,x_n\}</math> is closed. To see why, observe that <math>T^c = \Big[\bigcup\{x_i\}\Big]^c = \bigcap\{x_i\}^c</math> is open, so <math>T</math> is closed.
| |
| − | #Closed intervals [a,b] are closed.
| |
| − | #'''Cantor Set''' Consider the interval [0,1] and call it C<sub>0</sub>. Let A<sub>1</sub> be equal {0, <math>\tfrac{2}{3}</math>} and let d<sub>n</sub> = <math>(\tfrac{1}{3})^{n}</math>. Let A<sub>n+1</sub> be equal to the set A<sub>n</sub>∪{x|x=a+2d<sub>n</sub>, a∈A<sub>n</sub>}. Let C<sub>n</sub> be <math>\textstyle \bigcup_{a\in A_n}</math>{[a,a+d<sub>n</sub>]}, which is the finite union of closed sets, and is thus closed. Then the intersection <math>\textstyle \bigcap_{i=1}^\infty {C_i}</math> is called the Cantor set and is closed.
| |
| − | | |
| − | === Exercises ===
| |
| − | # Prove that a point x has a sequence of points within X converging to x if and only if all balls containing x contain at least one element within X.
| |
| − | # In <math>\mathbb{R}</math> the only sets that are both open and closed are the empty set, and the entire set. This is not the case when you look at <math>\mathbb{Q}\subseteq\mathbb{R}</math>. Give an example of a set which is both open and closed in <math>\mathbb{Q}</math>.
| |
| − | # Let <math>A</math> be a set in the space <math>x</math>. Prove the following:
| |
| − | ## <math>Cl(A) = Int(A^c)^c</math>
| |
| − | ## <math>Int(A) = Cl(A^c)^c</math>
| |
| − | | |
| − | ==Continuity==
| |
| − | ===Definition ===
| |
| − | Let's recall the idea of continuity of functions. Continuity means, intuitively, that you can draw a function on a paper, without lifting your pen from it. Continuity is important in topology. But let's start in the beginning:
| |
| − | | |
| − | The classic delta-epsilon definition: Let <math>(X,d),(Y,e)</math> be spaces. A function <math>f : X \rightarrow Y </math> is '''continuous''' at a point <math>x</math> if for all <math>\epsilon_x > 0</math> there exists a <math>\delta_{\epsilon_x} > 0</math> such that:
| |
| − | for all <math>x_1</math> such that <math>d(x,x_1) < \delta_{\epsilon_x} </math>, we have that <math>e(f(x), f(x_1)) < \epsilon_x</math>.
| |
| − | | |
| − | Let's rephrase the definition to use balls: A function <math>f : X \rightarrow Y </math> is continuous at a point <math>x</math> if for all <math>\epsilon_x > 0</math> there exists <math>\delta_{\epsilon_x} > 0</math> such that the following holds:
| |
| − | for every <math>x_1</math> such that <math>x_1 \in B_{\delta_{\epsilon_x}} (x) </math> we have that <math> f(x_1) \in B_{\epsilon_{x}}(f(x))</math>. Or more simply:
| |
| − | <math> f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))</math>
| |
| − | | |
| − | Looks better already! But we can do more.
| |
| − | | |
| − | '''Definitions:'''
| |
| − | *A function is continuous in a set S if it is continuous at every point in S.
| |
| − | *A function is continuous if it is continuous in its entire domain.
| |
| − | | |
| − | '''Proposition:'''
| |
| − | A function <math>f : X \rightarrow Y </math> is continuous, by the definition above <math>\Leftrightarrow</math> for every open set <math>U</math> in <math>Y</math>, The inverse image of <math>U</math>, <math>f^{-1}(U)</math>, is open in <math>X</math>. That is, the inverse image of every open set in <math>Y</math> is open in <math>X</math>.<BR/>
| |
| − | Note that <math>f</math> does not have to be surjective or bijective for <math>f^{-1}</math> to be well defined. The notation <math>f^{-1}</math> simply means <math>f^{-1}(U) = \{x \in X: f(x) \in U\}</math>.
| |
| − | | |
| − | '''Proof:'''
| |
| − | First, let's assume that a function <math>f</math> is continuous by definition (The <math>\Rightarrow</math> direction). We need to show that for every open set <math>U</math>, <math>f^{-1}(U)</math> is open.
| |
| − | | |
| − | Let <math>U\subseteq Y</math> be an open set. Let <math>x \in f^{-1}(U)</math>.
| |
| − | <math>f(x)</math> is in <math>U</math> and because <math>U</math> is open, we can find and <math>\epsilon_x</math>, such that <math>B_{\epsilon_x}(f(x)) \subseteq U</math>. Because f is continuous, for that <math>\epsilon_x</math>, we can find a <math>\delta_{\epsilon_x} > 0</math> such that <math> f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x)) \subseteq U</math>. that means that <math>B_{\delta_{\epsilon_x}}(x) \subseteq f^{-1}(U)</math>, and therefore, <math>x</math> is an internal point. This is true for every <math>x</math> - meaning that all the points in <math>f^{-1}(U)</math> are internal points, and by definition, <math>f^{-1}(U)</math> is open.
| |
| − | | |
| − | (<math>\Leftarrow</math>)<U>On the other hand</U>, let's assume that for a function <math>f</math> for every open set <math>U \in Y</math>, <math>f^{-1}(U)</math> is open in <math>X</math>. We need to show that <math>f</math> is continuous.
| |
| − | | |
| − | For every <math>x\in X</math> and for every <math>\epsilon_x > 0</math>, The set <math>B_{\epsilon_x}(f(x))</math> is open in <math>Y</math>. Therefore the set <math>V = f^{-1}(B_{\epsilon_x}(f(x)))</math> is open in <math>X</math>. Note that <math>x\in V</math>. Because <math>V</math> is open, that means that we can find a <math>\delta_{\epsilon_x}</math> such that <math>B_{\delta_{\epsilon_x}}(x) \subseteq V</math>, and we have that <math> f(B_{\delta_{\epsilon_x}}(x)) \subseteq B_{\epsilon_{x}}(f(x))</math>.
| |
| − | | |
| − | The last proof gave us '''an additional definition we will use for continuity for the rest of this book'''. The beauty of this new definition is that it only uses open-sets, and there for can be applied to spaces without a metric, so we now have two equivalent definitions which we can use for continuity.
| |
| − | | |
| − | === Examples ===
| |
| − | * Let <math>f</math> be any function from any space <math>(X,d)</math>, to any space <math>(Y,e)</math>, were <math>d</math> is the discrete metric. Then <math>f</math> is continuous. Why? For every open set <math>U</math>, the set <math>f^{-1}(U)</math> is open, because every set is open in a space with the discrete metric.
| |
| − | * Let <math>f:\mathbb{R}\rightarrow\mathbb{R}; f(x)=x</math> The identity function. <math>f</math> is continuous: The source of every open set is itself, and therefore open.
| |
| − | === Exercise ===
| |
| − | # Prove that a function <math>f : X \rightarrow Y </math> is continuous <math>\Leftrightarrow</math> for every closed set <math>U</math> in <math>Y</math>, The inverse image of <math>U</math>, <math>f^{-1}(U)</math>, is closed in <math>X</math>. <BR/>
| |
| − | | |
| − | === Uniform Continuity ===
| |
| − | In a metric space X, function from X to a metric space Y is '''uniformly continuous''' if for all <math>\epsilon</math>, there exists a <math>\delta</math> such that for all <math>x_1,x_2\in X</math>, <math>d(x_1,x_2)<\delta</math> implies that <math>d(f(x_1),f(x_2))<\epsilon</math>.
| |
| | | | |
| | == Licensing == | | == Licensing == |
| | Content obtained and/or adapted from: | | Content obtained and/or adapted from: |
| − | * [https://en.wikibooks.org/wiki/Topology/Metric_Spaces Metric Spaces, Wikibooks: Topology] under a CC BY-SA license | + | * [https://en.wikipedia.org/wiki/Metric_space#Types_of_maps_between_metric_spaces Metric Space, Wikipedia] under a CC BY-SA license |
![{\displaystyle f^{-1}[U]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c2decf23632b393104f53795355291a1224daa4)
