Can Infinite join exists in Heyting algebra?

Question

I couldn't be able to answer this question. A Heyting algebra $(A,\vee,\wedge,\rightarrow,0,1)$ primarily a bounded distributive lattice with top element 1 and bottom element 0. so what about the join of any subset of $A$. Please help..

Answer 1

Every bounded totally ordered set is a Heyting Algebra.
Not every bounded totally ordered set is a complete lattice, and thus it doesn't have arbitrary joins.
For example, take $C = \{-1/n : n \in \mathbb N\} \cup \{ 1/n : n \in \mathbb N\}$, with the order inherited from $\mathbb Q$.
It is a bounded totally ordered set, with least element $-1$ and greatest element $1$, whence a Heyting Algebra.
However, the set $\{x \in C : x < 0\}$ doesn't have a join.

For more on this topic, see Complete Heyting Algebra.