Your model includes referential cycles. This comes up quite a bit when you're modeling and you haven't worked through the differences between what relationships exist and what relationships are important to your system.
Without knowing more about your business rules and their justification, I would suggest that you consider dropping teachers.topic_id and possibly also teachers.subject_id.
Your lessons table relates subjects and topics. Since you have a many-to-many relationship between teachers and lessons, you can use lessons (via teachers_and_lessons) to tell you which teachers can teach which topics and which subjects.
I doubt that neither teachers.topic_id nor teachers.subject_id are correct anyway, since this would imply that any given teacher can only teach one topic and one subject. I don't know about your school, but it sounds much too specialized to me!
Edit: Recognizing Teaching Qualifications
Based on a comment by OP (below) the question is how can a teacher be recognized as being qualified to teach a topic without there being an actual lesson assignment made (yet)?
Issue 1: The relationship of topics to subjects:
It isn't clear from the original question whether topics belong to one subject or possibly to multiple subjects. If topics (e.g. fractions) belong to only one subject (e.g. math) then there should be a foreign key like so: topics.subject_id. If this were so, then lessons.subject_id would be redundant and could also be dropped. This would simplify the model and make things much clearer while reducing the risk of inconsistent foreign keys.
Issue 2: What teachers can do vs. what they are doing:
The teachers_and_lessons table shows which lessons teachers are actually teaching. Before this assignment is made, OP would like to be able to record which topics a teacher is qualified to teach. This could be done with a second M:N intersection, like so:
teacher_qualifications:
(teacher_id (PK/FK), topic_id (PK/FK))
This design assumes the simplification suggested under Issue 1, above. One potential drawback of this simplistic solution is that there is no constraint against assigning a teacher to teach a lesson for which they are unqualified. This constraint, if it is important, could be accommodated by replacing the teacher_qualifications and the teachers_and_lessons tables with new tables as follows:
teacher_qualifications:
(qualification_id (PK), teacher_id (FK,UN1), topic_id (FK,UN1))
lesson_instructor:
(qualification_id (PK,FK), lessons_id (PK,FK))
Again, these tables assume the simplification from Issue 1. Even without that simplification, these tables could be amended to use an analogous approach.
Best Answer
My recommendation would be to have the subject table with just subject_name and subject_id, have the grades table with just the grade_name and grade_id. Then create a third table called subject_grades that has the columns subject_id and grade_id, on this table create foreign keys to the subject.subject_id column and the grades.grades_id column. I'd highly recommend putting a primary key over both of the columns in this table or at least a unique key.
I hope this helps you.