Newbetuts

There is a set of continuous functions $f$ on [0, 1] with supremum metric (metric space). Proof that $\phi(f) = f(0) + f(1)$ is continous

How to finish this proof about compact implies bounded

Show that the discrete topology on $X$ is induced by the discrete metric

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Motivation of generalizing the theory of metric spaces to the theory of topological spaces

Finite union of compact sets is compact

Are there any different proof of the uncountability of $[0, 1]$?

How is $xy=1$ closed in $\Bbb{R}^2$?

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).

Metric on $\Bbb{R}$

Prove that a compact metric space can be covered by open balls that don't overlap too much.

$f:X \to Y $ is continuous on $X$ and $(X, d_1) $ is compact. Then $f:X\to Y$ is uniformly continuous on $X$

Show that any two closed and bounded interval are homeomorphic in $\mathbb{R}$

Where i am wrong? A question on uniformly continuous function in functional analysis.

Triangle inequality for the distance between two sets

New posts in metric-spaces

Distance between two points in different sets

Canonical metric on product of two complete metric spaces

What is meant by gluing two metric spaces together?

What is a mathematical definition of the Maxwellian spacetime?

There is a set of continuous functions $f$ on [0, 1] with supremum metric (metric space). Proof that $\phi(f) = f(0) + f(1)$ is continous

How to finish this proof about compact implies bounded

Show that the discrete topology on $X$ is induced by the discrete metric

If every real valued continuous function on $X$ is uniformly continuous , then is every continuous function to any metric space uniformly continuous?

Example of two open balls such that the one with the smaller radius contains the one with the larger radius.

Motivation of generalizing the theory of metric spaces to the theory of topological spaces

Finite union of compact sets is compact

Are there any different proof of the uncountability of $[0, 1]$?

How is $xy=1$ closed in $\Bbb{R}^2$?

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).

Metric on $\Bbb{R}$

Prove that a compact metric space can be covered by open balls that don't overlap too much.

$f:X \to Y $ is continuous on $X$ and $(X, d_1) $ is compact. Then $f:X\to Y$ is uniformly continuous on $X$

Show that any two closed and bounded interval are homeomorphic in $\mathbb{R}$

Where i am wrong? A question on uniformly continuous function in functional analysis.

Triangle inequality for the distance between two sets