Let $\beta$ be the collection of all open intervals $(a,b) \subset \mathbb{R}$, with $a,b \in \mathbb{R}$. Prove that the topology $\tau_\beta$ generated by $\beta$ is in fact a topology. Observation:

$\tau_\beta = \{U \subset \mathbb{R} \ \vert \ \forall x \in U ,\exists B \in \beta \text{ such that } x \in B \subset U\}$

My proof: $\emptyset \in \tau_\beta$ vacuously. Since for every $x \in \mathbb{R}$ and $\varepsilon > 0$, we have that $x \in (x-\varepsilon, x+\varepsilon) \subset \mathbb{R}$, then $\mathbb{R} \in \tau_\beta$. Now, given any arbitrary $x \in A = \displaystyle{\bigcup_{j \in J} U_j} $, where $U_j$ is an open set for each $j$, then $x \in U_i$ for some $i \in J$. Since $U_i $ is open, then there exists an interval $B$ such that $x \in B \subset U_i$. Finally, consider $W = U_1 \bigcap U_2$, where $U_1$ and $U_2$ are open sets. Given any $x \in W$, since $x \in U_1$ and $x \in U_2$, then there exist $B_1, B_2 \in \beta$ such that $x \in B_1 \subset U_1$ and $x \in B_2 \subset U_2$. Then $x \in B_1 \bigcap B_2 \subset U_1 \bigcap U_2$ and we're done (the rest is trivial by induction).

Are all my steps correct?

  • 1
    $\begingroup$ Yes it's correct. Maybe you should spell out the observation in the beginning.. $\endgroup$ – Berci Jan 7 at 22:48
  • $\begingroup$ Isn't the topology generated by a subset of $P(A)$ always a topology on $A$? (See @William Elliot's answer). Out of curiosity, where is this problem from? $\endgroup$ – Chris Custer Jan 8 at 3:42
  • $\begingroup$ This was an exercise given on my first day of a general topology summer course. $\endgroup$ – Matheus Andrade Jan 8 at 10:15
  • $\begingroup$ Ok. I guess it gets you to work through the definitions. $\endgroup$ – Chris Custer Jan 8 at 19:31

That is a vacuous problem because the topology generated by any subset K of P(X) is by definition a topology for X, the smallest topology for X containing K, the intersection of all topologies for X containing K.

Exercise. If T is a not empty collection of topologies for X,
show $\cap$T is a topology for X.

Problem. Is there a topology for X that contains any subset of P(X)?
Why is this important?


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