What are some counter-intuitive results that may occur in metric space but not in $\mathbb R$.

Question

I have just started studying metric space and I have studied upto open sets in metric spaces and open balls and know some examples of standard metric spaces.I have seen many counter-intuitive things happening in certain metric space for example a smaller ball may properly contain a larger ball,Closure of an open ball of a fixed radius $r$ and with centre $x$ may not be equal to the corresponding closed ball etc.I am looking for some more examples of such weird stuff and I want to find examples of metric spaces where those things happen in order to strengthen my intuition and ability of constructing examples.

Can anyone provide me with some more such results involving open sets.(Notice again that I have yet not learnt compactness,sequences,completeness etc in metric spaces,I have only started with open sets,interior,closure etc.,bounded metric etc,diameter of a set,distance between sets).

Answer 1

We spend a lot of time in real analysis telling students that a set being not open does not mean it is necessarily closed, and a set being not closed does not mean it is necessarily open. But we can have metric spaces which are door spaces, that is every set is either open or closed (but possibly both). Consider for example, the space defined on the real numbers by defined by $d(x,y)= \mathrm{max}{|x|,|y|}$ when $x \neq y$ and by $d(x,y)=0$ when $x=y$. This is a space with a lot of fun things in general.

If you've seen p-adic metrics on the rationals or integers, that's also a good source of some metrics which make spaces behave very differently than the way one is used to on the reals.

Answer 2

• You may have sets which are both closed and open but which are neither $\emptyset$ nor the whole space.
• You may have Cauchy sequences which do not converge.
• You may have uncountable sets such that the distance between any two distinct points is greater than a fixed constant.