About finding the closure of a set.

Question

I want to learn how should I find the closure of a set $S$. It's a long that I was not worked with topological concepts, and I am afraid if I am mistaken about some easy things. Here I will list some of the facts which will help me to solve my problems and pursue my purposes.

Suppose that a topological space $X$ is given, and let $R, S \subseteq X$ be two sets.

1. Is this true? The closure of $S$ is the union of $S$ with all limit points of $S$. (I think it should be the definition of the closure, or at least it should be equivalent to the definition, any other equivalent definitions are welcome.)
2. If $S$ is dense in $R$, then $R \subseteq \overline S$.
3. $\overline S$ is the largest set, in which $S$ is dense. (i.e. if $\overline S \subsetneq R'$, then any $x\in R'\backslash \overline S$ has a positive distance from $S$, therefore is not a limit point)

The next two points, are not related to the closure, but I have some doubts.

4. A function $f: X \rightarrow Y$ between two topological spaces is called continuous, iff for every open $V \subseteq Y$, $f^{-1}(V)$ is open in $X$. (I think this was a standard definition, any other equivalent definition are welcome.)
5. A function $f: X \rightarrow Y$ between two topological spaces, is continuous, iff for every close $V \subseteq Y$, $f^{-1}(V)$ is close in $X$.

My intuition says that, the first $4$ points are true. Am I right? If I am mistaken about these facts, please tell me, and if it is possible please give me a counter-example. Also, I think the last point is also equivalent to the $4^{th}$ point, because if a set is closed iff and only if its complement is open. Having this in mind it seems the last two points are equivalent to each other as the definition of a continuous function.

Any equivalent definitions to the $1^{st}$ point and $4^{th}$ point are welcome. Especially those equivalent definitions, which are useful for computing the closure. Using the first three points I am able to find the closure of some simple sets in $p$-adic topology.

Answer 1

1. Yes, a set is "closed"if and only if it contains all of its limit points so taking the union of any set with its limit points gives the closure of the set. (The closure of a set is also the intersection of all closed sets containing it.)

2.Yes, that is pretty much the definition of "dense".

3. Yes, again that follows directly from the definition of "dense".

4. The spelling is "continuous", not "continues". But, yes, that is a standard definition of "continuous".

5. Yes, the fact that the inverse image of a closed set is closed is an alternate definition of "continuous