Recent posts

Lebesgue Integration

3 minute read

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...

Dimension Theorem and Rank Theorem

3 minute read

Definition Let $A\in \mathfrak{M}_{m\times n}(\mathbb{R})$ be a matrix. We define the column rank of $A$ as dimension of $\langle [A]^1, \ldots, [A]^n \rangl...

Closed Sets and Limit Points

1 minute read

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

Product Topology and Subspace Topology

4 minute read

Definition (Product Topology on $X \times Y$) Let $(X,\mathfrak{T}_X)$ and $(Y,\mathfrak{T}_Y)$ be topological spaces. The product space topology on $X\times...

Topology and Basis

5 minute read

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