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## Homework Statement

Hi everybody! I'd like to check with you guys if I tackled that problem correctly. I might have a few theoretical questions along the way :)

Prove that the difference ##A \setminus B## of a closed set ##A \subset \mathbb{R}^2## and an open set ##B \subset \mathbb{R}^2## is a closed set.

## Homework Equations

First I'd like to define open/closed sets in ##\mathbb{R}^2##:

- a set ##M_1 \subset \mathbb{R}^2## is called open, if none of its boundary points is included in the set;

- a set ##M_2 \subset \mathbb{R}^2## is called closed, if it contains all of its boundary points.

I will use also the following theorems:

1. If ##X## is a topological space and ##U## is a subset of ##X##, then the set ##U## is called closed when its complement ##X \setminus U## is an open set.

2. The intersection of two closed sets is a closed set.

## The Attempt at a Solution

My first question concerns the topological space. I have read numerous articles (mostly Wikipedia I admit, but not only) about what a topological space is, and I still don't get what it really is. Neither did I understand any of the few examples I saw... I used the theorem nevertheless, assuming ##\mathbb{R}^2## is a topological space, and if not I theorised it may be a subset of ##\mathbb{R}^2## containing both ##A## and ##B##.

Applying the 1st theorem, I can develop as follows:

Let ##X = \mathbb{R}^2## be a topological space (...or not).

##B## is an open set ##\implies## its complement ##B^c = X \setminus B## is closed

##\implies\ A \setminus B = A \cap B^c## is closed, because of theorem 2 (##A## and ##X \setminus B## are closed sets).

Is that a valid proof? Do the general definitions of open/closed sets hold in ##\mathbb{R}^n##?

Thanks a lot in advance for your answers, I appreciate your help!

Julien.