I'm trying to show that the interval $(0,1)$ is uncountable and I want to verify that my proof is correct My solution: Suppose by way of contradiction that $(0, 1)$ is countable. Then we can …

Nov 5, 2015 · My friend and I were discussing infinity and stuff about it and ran into some disagreements regarding countable and uncountable infinity. As far as I understand, the list of …

The question is not well-posed because the notion of an infinite sum $\sum_ {\alpha\in A}x_\alpha$ over an uncountable collection has not been defined. The "infinite sums" familiar …

However, it is perhaps more common that we first establish the fact that $ (0, 1)$ is uncountable (by Cantor's diagonalization argument), and then use the above method (finding a bijection …

A space for which any two points can be separated by a continuous real-valued function is said to be functionally Hausdorff and if it's connected and has more than two points, it's uncountable.

Jun 24, 2018 · Countable. Uncountable. Countable or uncountable. Finite. At most countable. I understand it has to be countable, that is option 3. But the answer given was option 7 - "at …

A set $A$ is countable if $A\approx\mathbb {N}$, and uncountable if it is neither finite nor countably infinite.

Sep 13, 2018 · In set theory, when we talk about the cardinality of a set we have notions of finite, countable and uncountably infinite sets. Main Question Let's talk about the dimension of a …

May 16, 2024 · We know the interval $ [0, 1]$ is uncountable. You can think of the binary expansions of all numbers in $ [0, 1]$. This will give you an uncountable collection of sequences.

Is it true that for any uncountable subset T of $\\mathbb R$, one can find a subset S of T such that S is countable. If yes, how can we prove it? Thanks! Edit: Is there a countable subset S of T s...