1

Following task in a textbook:

Give a characterization of the subspace topology through:

a) closed sets

b) closure

c) neighborhoods

And I am not sure what actually is asked here. So what a suitable characterization should fulfill. Of course it should be equivalent.

So let $(X,\tau)$ be a topological space, and $(Y,\tau_Y)$ a subspace with the subspace topology. For a) I thought about this at first. Take $\{A\subseteq Y: A=Y\cap U^c~\text{with $U\in\tau$}\}$.

As $U^c$ is closed, this (somewhat) gives a characterization with closed sets. But this was not satisfying at all.

It is easy to show that a set $A\subseteq Y$ is closed (in $Y$) iff it exists a closed set (in $X$) $W\subseteq X$ with $A=Y\cap W$.

Which seems like a more suitable answer, and I could discribe the subspace topology in terms of closed sets in the obvious way.

But what about b)?

I know that if $A\subseteq Y$ then $\overline{A}\cap Y$ is the closure of $A$ in $Y$. But how would that lead to a characterization of the subspace topology?

It feels more like I am giving a characterization of the terms closed set, closure, neighborhood with regards to the subspace topology.

So I would say that $U\subseteq Y$ is a neighborhood of $y\in Y$ (with regards to the subspace topology), iff there is a neighborhood $V\subseteq X$ of $y$ with $U=Y\cap V$.

But I am not sure about the solution the task wants you to find. Can you help me out. Are my thoughts correct?

Thanks in advance.

Cornman
  • 11,065
  • 4
  • 30
  • 57
  • Thing is, a topology can be defined using closures. Try to google "Kuratowski's closure axioms" and you will find it. So I guess here you are asked to describe the closure operator of the subspace topology. (which uniquely defines the topology) – Mark Dec 31 '20 at 00:41
  • @Mark Thanks for the keyword. I am familiar with the Kuratowski closure, as it came up before. I have to think about it. What do you think about my thoughts anyways? Or how can I define the topology with regards to neighborhoods? – Cornman Dec 31 '20 at 02:17
  • Which textbook do you use? – Paul Frost Dec 31 '20 at 07:03
  • @Cornman Well, you already wrote what is the closure of each set. So now I believe you just have to check that these closures satisfy the Kuratowski axioms (using the fact that the closures in the original space satisfy them. This shouldn't be difficult), and that gives you a characterization of the topology. As for neighborhoods, again, a topology can be defined by defining the neighborhood of each point. For example, see here: https://math.stackexchange.com/questions/2692678/why-does-the-definition-of-a-topology-via-neighborhoods-include-this-axiom – Mark Dec 31 '20 at 09:01
  • @PaulFrost It is a german textbook called "Grundkurs Topologie" by Gerd Laures and Markus Szymik. – Cornman Dec 31 '20 at 09:52
  • @Mark Thank you. I will take a closer look at it, and might present further details later. – Cornman Dec 31 '20 at 09:52
  • 1
    I believe you are overthinking this exercise. It is not after some deep theory. Rather its point is to drive home the many various ways to define a topology, but having you use those same ways to define the supspace topology. A topology is completely determined by (1) what sets are open under it, or by (2) what sets are closed under it, or by (3) characterizing what the closure of any set in it is, or by (4) characterizing what constitutes neighborhoods of its points. (Or by many other means not mentioned here). The answers you give do not have to be deep or complicated. – Paul Sinclair Dec 31 '20 at 14:23

1 Answers1

2

Possible answers (proofs left to the reader, and not hard)

If $Y \subseteq X$ is a subset of a space $X$, then the closed sets of the subspace topology on $Y$ are exactly of the form $C \cap Y$ where $C$ is closed in $X$.

If $A \subseteq Y$ we can define the subspace closure operation in terms of the closure on $X$ by $$\operatorname{Cl}_Y(A) = \operatorname{Cl}_X(A) \cap Y$$

And the neighbourhood system for $y \in Y$ in the subspace topology is given by $$\mathcal{N}_Y(y) = \{N \cap Y: N \in \mathcal{N}_X(y)\}$$

All of these characterise the subspace topology. You can define a topology by giving its open sets, its closed sets, its closure operation or its neighbourhood systems (there are other ways too). The above characterisations follow quite easily from knowing what the open sets are and how closure and niieghbourhood systems are defined in terms of open sets.

Henno Brandsma
  • 242,131