Suppose that we have a set $S$ containing 0 and 1. Can we define our topology to be the four open sets $\varnothing$, $\{0\}$, $\{1\}$ and $\{0,1\}$? I know that the Sierpinski set contains the three elements $\varnothing$, $\{0\}$, and $\{0,1\}$. I wonder if the fourth element is added, is it still a topology.
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Can someone just explain to me the basic process of what is going on here? I understand everything until we start adding 1's then after ...
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