Thread:

1. Basic Topology, closed and open sets

Consider the metric space ([0, 1], | · |).In this space, no number smaller than 0 or more than 1 exists.

a) Show that any interval in the form (r,s), where 0 < r < s < 1 is open in this space.

b)Show that any interval in the form [0, s), where s < 1, is open in this space.

c) Show that any interval in the form (r, 1], where r > 0, is open in this space.

2. Hello, there may be a simpler way, but this proved it for me:

a) Let $y\in (r,s)$. Then, for $\delta =\min \{d(r,y),d(s,y)\},$ we have that $B_d(y,\delta )\subset (r,s)$, hence $(r,s)$ is open.

For b) and c), adapt the same argument accordingly.
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