Hey Please help me,I have some doubts about basic topology concepts concepts

how can I help?

@LeakyNun See this image

This definition looks like saying every interior point is adherent point

I would say $((a-\delta,a+\delta)\setminus\{a\}) \cap S \ne \varnothing$ instead, i.e. exclude the point itself

@MohanSharma either you didn't mean "interior" or you're wrong (even if you assume that definition)

hmm, interesting definition of adherent points

wait, that's correct

@LeakyNun in Q why we can not find an open interval around any rational number

this one is for limit point

@MohanSharma because every open interval contains an irrational number

uncountably many, in fact

If A is set and Int(A) is empty set .What we can say about A

@LeakyNun If A is set and Int(A) is empty set .What we can say about A

how do I know

@LeakyNun Looking at Wikipedia it seems that (at least some authors) distinguish between adherent point and limit point.

@MartinSleziak yes, indeed

Yes even i don't know

> In mathematics, an adherent point (also closure point or point of closure or contact point)of a subset A of a topological space X, is a point x in X such that every open set containing x contains at least one point of A. A point x is an adherent point for A if and only if x is in the closure of A.

@MohanSharma Its complement is dense. Which is why such set is sometimes called co-dense or boundary set.

Book says Int (A) is empty

Yes, it definitely is.

All points of $A$ are isolated.

But A=(0,1) why Int(A) is not equal to A

@MohanSharma what

how is A=(0,1)

This post asks about interior of that set: Interior points/Boundary points of sets

And also this one: Interior and closure of $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} $ in $\mathbb{R}$ and $\mathbb{R} \setminus \{ 0 \}$

@MohanSharma bear in mind, $n \in \Bbb N$

Soory it is (0,1]

I mean A=(0,1]

no it isn't

Okk yes n is natural number not real number

@LeakyNun Thanks for answering all my stupid question.May i ask you help also in future