Khanh at 22:57, 13 November 2021

2021-11-13T22:57:33Z

Khanh: Created page with "==Metric Space== ===Definition=== A metric space is a Cartesian pair $(X,d)$ where $X$ is a non-empty set and $d:X\times X\to[0,\infty)$ , is a..."

2021-11-13T22:55:11Z

Created page with "==Metric Space== ===Definition=== A metric space is a Cartesian pair <math>(X,d)</math> where <math>X</math> is a non-empty set and <math>d:X\times X\to[0,\infty)</math>, is a..."

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@@ Line 43: / Line 43: @@
 ===Examples===
-Why is this called a [[w:Ball_(mathematics)|ball]]? Let's look at the case of <math>\R^3</math>:
+Why is this called a ball? Let's look at the case of <math>\R^3</math>:
 :<math>d\bigl((x_1,x_2,x_3),(y_1,y_2,y_3)\bigr)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+(x_3-y_3)^2}</math>
 Therefore <math>B_r\bigl((0,0,0)\bigr)</math> is exactly <math>x_1^2+x_2^2+x_3^2<r^2</math> – The ball with <math>(0,0,0)</math> at center, of radius <math>r</math>. In <math>\R^3</math> the ball is called ''open'', because it does not contain the sphere (<math>x_1^2+x_2^2+x_3^2=r^2</math>).
@@ Line 295: / Line 295: @@
 '''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 [[w:Inverse image|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/>
+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>.

A Topology Given By A Metric - Revision history

Khanh at 22:57, 13 November 2021

Khanh: Created page with "==Metric Space== ===Definition=== A metric space is a Cartesian pair (X,d) where X is a non-empty set and d:X\times X\to[0,\infty), is a..."

Khanh: Created page with "==Metric Space== ===Definition=== A metric space is a Cartesian pair $(X,d)$ where $X$ is a non-empty set and $d:X\times X\to[0,\infty)$ , is a..."