Consider a set equiped with two topologies. What does it mean to say that the two topologies coincide at a point in the set? Is it meaningful to talk about this concept in general. Is it meaningful in the context of metric spaces?

p.s. I know what the relative topology is for any subset of a topological space and this concept is a different one, as is apparent from this problem.

  • 1
    $\begingroup$ Is this not coming from a book you are studying? And the context is not included in that book? $\endgroup$ – Jonas Meyer Jul 9 '13 at 6:49
  • $\begingroup$ With no context I’d take it to mean that if $p$ is the point and $\tau_1$ and $\tau_2$ are the topologies, then for each $A\subseteq X$, $x\in\operatorname{cl}_{\tau_1}A$ iff $x\in\operatorname{cl}_{\tau_2}A$. That, however, would be a guess. $\endgroup$ – Brian M. Scott Jul 9 '13 at 6:51
  • $\begingroup$ It really depends on your book, but I have provided two decent guesses below. $\endgroup$ – Thomas Andrews Jul 9 '13 at 7:11
  • $\begingroup$ I think @BrianM.Scott's definition is equivalent to my first definition. $\endgroup$ – Thomas Andrews Jul 9 '13 at 7:22
  • $\begingroup$ @Thomas: Yes; they both amount to saying that the point has the same nbhds in both topologies, where by nbhd I mean a set that contains the point in its interior. $\endgroup$ – Brian M. Scott Jul 9 '13 at 9:11

I would guess that it means the every open neighborhood of $x$ in $\tau_1$ contains an open neighborhood of $x$ in $\tau_2$, and visa versa. This essentially means that "continuity at $x$" is the same in each topology.

It's possible, though, that the stronger statement is intended: the neighborhoods of $x$ are the same in each topology.

There's a lovely way to define the "local topology" at a point. Let $(X,\tau)$ be a topology and $x\in X$. Define a new topology, $\tau_x$ which has as a basis the singletons $\{y\}$ where $y\neq x$ and $U\in\tau$ such that $x\in U$. So the open sets are any set not containing $x$, or any set contain a $\tau$-neighborhood of $x$. It is not hard to see that this is a topology.

In this case, my first definition would be equivalent to saying, for topologies $\tau$ and $\rho$, they coincide at $x$ if and only if $\tau_x=\rho_x$.

Now, in the complete lattice of topologies on $X$, $$\bigcap_{x\in X}\tau_x = \tau$$ So we can recover $\tau$ if we know each $\tau_x$. Then, for a topological space, $Y$, we can define a function $f:X\to Y$ to be $\tau$-continuous at $x$ if it is continuous on $(X,\tau_x)$, and it turns out, due to the lattice meet property above, the function $f$ is continuous on $(X,\tau)$ if and only if it is continuous at each $(X,\tau_x)$.

There's also a simple "local range topology", $\tau^x$ defined as just the $\tau$-open sets containing $x$, and the empty set. Then you can say that $f:Y\to X$ is continuous "to" $x$ if it is continuous when using the topology $\tau^x$. Again, we can recover $\tau$ if we know ecah $\tau^x$. Indeed, $\tau = \bigcup \tau^x$.

The stronger definition above would happen exactly when $\tau^x=\rho^x$. It's true that if $\tau^x=\rho^x\implies\tau_x=\rho_x$, though so maybe this stronger definition is reasonable.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.