For instance, what does it mean to say that the lower limit topology on $\mathbb{R}$ is strictly finer than the usual topology on $\mathbb{R}$?

I understand why lower limit topology is finer.

Take any open $(a,b), b>a$ in $\mathbb{R}_{usual}$, we can represent $(a,b) = \bigcup\limits_{n \in \mathbb{N}} [a+\dfrac{\epsilon}{n}, b)$, where $\epsilon < \dfrac{b-a}{2}$

Then all the sets in $\tau_{usual} \subseteq \tau_{ll}$.

  1. What does it mean for the lower limit topology to be strictly finer?

  2. Are there two topologies such that one is finer but not strictly finer?

  • 1
    $\begingroup$ Every topology is finer than itself, but not strictly finer than itself. $\endgroup$ – Carl Mummert Jun 14 '16 at 2:02
  • $\begingroup$ As it happens with all the ordering relations, "strictly finer" means "finer and not equal". $\endgroup$ – user228113 Jun 14 '16 at 2:07
  • $\begingroup$ @G.Sassatelli Then is it ever possible for two different topologies where one is non-strictly finer to each other? $\endgroup$ – Olórin Jun 14 '16 at 2:09
  • $\begingroup$ It's not possible. $\endgroup$ – user228113 Jun 14 '16 at 2:11

Summarising the comments:

  • The lower limit topology being strictly finer than $\tau_{\text{usual}}$ means it's not equal to $\tau_{\text{usual}}$.

  • It's essentially the difference between $<$ and $\le$: $x < y$ means exactly $x \le y$ and $x \neq y$. Or really, the difference between $\tau_1 \subsetneq \tau_2$ and $\tau_1 \subseteq \tau_2$.

  • The only way $\tau_1$ can be finer than $\tau_2$ but not strictly finer, is when $\tau_1 = \tau_2$.

  • It follows also that these two topologies cannot be strictly finer than each other, just like we cannot have $x < y$ and $x > y$ at the same time.

  • If $\tau_1$ is finer than $\tau_2$ and $\tau_2$ is finer than $\tau_1$ (no stricter), then we have $\tau_1 \subseteq \tau_2$ and $\tau_2 \subseteq \tau_1$, so $\tau_1 = \tau_2$.

So to see the strictness in your example, it suffices to notice that $[0,1)$ is open in the lower limit topology, but not in the usual topology, e.g.

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  • $\begingroup$ Thanks, better than my prof and the text book combined $\endgroup$ – Olórin Jun 14 '16 at 18:52
  • $\begingroup$ @MSEisadatingsite Glad you like it! $\endgroup$ – Henno Brandsma Jun 14 '16 at 18:53

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