Topology: Indiscrete/Discrete Topology

[jeff1evesque]

I am reading from my text, and was just wondering if someone could provide additional information on the following examples.

0.1 Examples. For any set X each of the following defines a topology for X.

(1) $$T_{*} = {A \subseteq X|a \in A \Rightarrow X \subseteq A},$$ Indiscrete Topology.

(2) $$T^{*} = {A \subseteq X|a \in A \Rightarrow {a} \subseteq A},$$ Discrete Topology.

Questions:
I was wondering how we can have the following statement (from above),
(1) $$a \in A \Rightarrow X \subseteq A$$
(2) $$a \in A \Rightarrow {a} \subseteq A$$

Thanks,JL

 

[n!kofeyn]

This is a really set-theoretic definition approach to defining these topologies, and they threw me off at first too. The indiscrete topology of X is the topology containing only the empty set and X itself. This agrees with their definition because if A=empty set, then nothing's in there so the implication to the right of the bar is "true" (I think some might say vacuously true, but I don't like the term). So A=empty set is in the topology. If X=A, then X is automatically a subset of A and thus in the topology. These are the only two subsets of X that satisfy their definition.

The discrete topology of X is just the collection of all subsets of X, i.e. the topology equals the power set of X. This again agrees with their definition because any subset A of X will satisfy the property that if $a\in A$, then $a\subseteq A$.

I think defining these topologies in this way and not explaining them is very poor writing. I don't see any reason why they would do so, because the definitions I gave (which was what I was taught and is in the book Topology by Munkres) are perfectly rigorous. Their definitions certainly give no immediate insight as to what the topologies actually consist of.