# Posts by Tag

## Topology and Basis

Definition (Topology) A topology on a set $X$ is a collection $\mathfrak{T}$ having the following properties: (1) $\emptyset$ and $X$ are in $\mathfrak{T}$ (...

## Open and closed set-(2)

Theorem 3.1.13 $U \subset \mathbb{R}$ is open $\Rightarrow$ $$\exists \{I_n\}$$: a finite or countable family of pairwise disjoint union of open intervals su...

## Open and closed set

Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...

## Closed Sets and Limit Points

Definition (Closed) A subset $A$ of a topological space $X$ is said to be closed if the set $X-A$ is open.

## Open and closed set-(2)

Theorem 3.1.13 $U \subset \mathbb{R}$ is open $\Rightarrow$ $$\exists \{I_n\}$$: a finite or countable family of pairwise disjoint union of open intervals su...

## Open and closed set

Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...

## Cauchy Sequence

Definition Let $$\{p_n\}_{n=1}^\infty$$ be a sequence in $\mathbb{R}$. The sequence is a Cauchy sequence if $\forall \epsilon >0, \exists N\in \mathbb{N}$...

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Open and closed set-(2)

Theorem 3.1.13 $U \subset \mathbb{R}$ is open $\Rightarrow$ $$\exists \{I_n\}$$: a finite or countable family of pairwise disjoint union of open intervals su...

## Open and closed set

Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...

## Closed Sets and Limit Points

Definition (Closed) A subset $A$ of a topological space $X$ is said to be closed if the set $X-A$ is open.

## Open and closed set

Definition 3.1.1 $E$ is a subset of $\mathbb{R}$. $p\in E$ is an interior point of $E$ if there is $\epsilon >0$ such that $N_{\epsilon} (p) \subset E$. ...

Lemma 1

## Compact set

Theorem 3.2.5 (a) $K$ is compact $\Rightarrow$ $K$ is closed and bounded

## Properties of the Riemann Integral

Theorem 6.1.9 Let $f:[a,b]\to\mathbb{R}$ be a bounded real valued Riemann integrable function with $Range f \subset [c,d]$. Let $\varphi:[c,d]\to\mathbb{R}$ ...

## Riemann Integral

Definition (upper sum, lower sum) Let $[a,b]$ with $a<b$ be a closed and bounded interval in $\mathbb{R}$. By a partition of $\mathscr{P}$ of $[a,b]$ we m...

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Set and Function

Lemma 1.1.0 (De Morgan’s law) (1) $(\cup_{i\in I} A_i)^c = \cap_{i\in I}A^c_i$

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Least Upper Bound Property

Definition (Ordered Set) Let $S$ be a set. An order on $S$ is a relation, denoted by $<$, with the following property

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For $\forall x,y\in \mathbb{R}, |x+y|\leq |x|+|y|$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For $\forall x,y\in \mathbb{R}, |x+y|\leq |x|+|y|$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For $\forall x,y\in \mathbb{R}, |x+y|\leq |x|+|y|$

## Sequences of real numbers

Theorem 2.1.3 (Triangular inequality) For $\forall x,y\in \mathbb{R}, |x+y|\leq |x|+|y|$

## Lebesgue Integration

Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...

Proposition 8.2.6 Let $\Omega$ be a measurable set, and let $f: \Omega\rightarrow [0,\infty]$ and $g: \Omega\rightarrow [0,\infty]$ be non-negative measurabl...