$\mathbb{R}^2 \rightarrow \mathbb{R}$

Prove that there does not exist an $1-1$ and continuous mapping from $\mathbb{R}^2$ to $\mathbb{R}$.

Solution

Necessarily discrete?

If the only convergent sequences in a space are eventually constant, is the space necessarily discrete?

Solution

Not a complete metric space

Prove that the set of the continuous functions $C[a, b]$ endowed with the metric

$$\rho(f, g) = \sqrt{\int_{a}^{b}\left | f(t)-g(t) \right |^2 \, {\rm d}t}$$

where $f, g \in C[a, b]$ is not a complete metric space.

Solution

Function preserving order

Examine if there exists an $1-1$ and onto function $f:\mathbb{Q} \rightarrow \mathbb{Q} \setminus \{0\}$ that preserves order in $\mathbb{Q}$?

Solution

Connected topological space

Let $\mathbb{X}$ be a topological space. Show that $\mathbb{X}$ is connected if and only if every continuous function $f:\mathbb{X} \rightarrow \{0, 1\}$ is constant.

Solution

Isometry in compact space

Let $(X, \rho)$ be a compact metric space and let $f:X \rightarrow X$ be an isometry. Prove that $f$ is onto. Based on the above fact , prove that the $\ell^2$ space (that is the space of all real sequences $x_n$ such that $\sum \limits_{n=1}^{\infty} x_n^2$ converges) is not compact under the metric $ \rho \left ( x_n, y_n \right )= \sqrt{\sum \limits_{n=1}^{\infty}\left ( x_n-y_n \right )^2}$

Solution

Partition and function

Let $E$ be a non empty set and $A, B$ be two non empty subsets of $E$. Consider the function $f:P(E) \rightarrow P(A) \times P(B)$ that is given by $f(X)=\left(X\cap A,X\cap B\right)$. Show that $\{A, B\}$ is a partition of $E$ if-f then function $f$ is $1-1$ and onto $P(A) \times P(B)$.

Solution

Dense subspace

Let $\left(X,||\cdot||\right)$ be an $\mathbb{R}$ normed space and $Y$ be a subspace of $\left(X,+,\cdot\right)$ such that , if $f \in X^*= \mathbb{B} (X, \mathbb{R})$ with $f|_{Y}=\mathbb{O}$ then $f=\mathbb{O}$. Prove that $Y$ is dense on $\left(X,||\cdot||\right)$.

Solution

Algebra and Topology

Let $(X, \mathbb{T})$ be a topological space. Consider the commutative ring with unity $\left(C(X,\mathbb{R}),+,\cdot\right)$ of all continuous functions $f: X \rightarrow \mathbb{R}$ . Prove that the topological space $(X, \mathbb{T})$ is connected if-f the ring $\left(C(X,\mathbb{R}),+,\cdot\right)$ is connected.

Solution