# Open subsets of a set

I am a bit confused about how an open set is defined.

In the book I am following it’s stated that an open subset is one which is in the topology on a set.

So for example, say we have a set $$X$$ and a topology $$T$$ on $$X$$, then for any U such that $$U\subset X$$ if $$U\in T$$ then U is open, then also the complement of U, U’ is closed. Okay makes sense

Now consider $$X = \{ 1, 2, 3 \}$$ and $$T = \{ \{1\}, \{\emptyset\}, \{1, 2, 3\}\}$$ It’s clear that T is a topology. Now consider the complement of {1}, {2, 3}, this set is then closed.

But if you define $$T = \{\{2, 3\}, \{\emptyset\}, \{1, 2, 3\}\}$$ which is also a topology, then the set {2, 3} is open and it’s complement {1} is closed.

So is it that under different topologies, certain subsets can be open and under others closed? Or is there something wrong with my logic

• "So is it that under different topologies, certain subsets can be open and others closed?" Exactly. Commented Aug 1 at 16:18
• You can even consider something like $\mathcal{T} = \{\emptyset, \{1, 2, 3\}, \{1\}, \{2, 3\}\}$, and both $\{1\}$ and $\{2, 3\}$ would be open and closed. Commented Aug 1 at 16:20
• "for any U such that $U\subset X$ if $U\in T$ then U is open" --- Note that the assumption $U\subset X$ is not needed here, since it is an immediate consequence of $U\in T.$ Commented Aug 1 at 16:23
• Off-topic but probably immediately relevant: Try not to think "subsets are open or closed". Try to think: "a topology places subsets on a spectrum with open subsets on one end and closed subsets on the other". Commented Aug 1 at 16:38

Depending on the topolgy that you define on a set, a non empty subset different to the total set could be open, closed, both, or neither.

A small example:

Let $$X = \{ 1, 2 \}$$, $$T_1 = \{\emptyset, X\}$$, $$T_2 = \{\emptyset, \{1\}, X\}$$, $$T_3 = \{\emptyset, \{2\}, X\}$$, $$T_4 = \{\emptyset, \{1\}, \{2\}, X\}$$.

It is clear that $$T_i$$ is a topology on $$X$$, $$\forall i = 1, 2, 3, 4$$.

In $$T_1$$, $$\{1\}$$ is neither open nor closed.

In $$T_2$$, $$\{1\}$$ is open but not closed.

In $$T_3$$, $$\{1\}$$ is closed but not open.

In $$T_4$$, $$\{1\}$$ is both open and closed.

It's worth considering two topologies when developing intuition about open sets.

Consider these: $$X\ne \emptyset$$ and with $$T_i=\{\emptyset,X\}$$ and $$T_f=\mathcal{P}(X)$$ ($$\mathcal{P}$$ denoting the power, as the topology).

Under $$T_i$$ only the empty set and the set $$X$$ are open, but they are also closed. Under $$T_f$$ every set is open, and every set is closed.

Any Topology $$T_i \subset T_\alpha \subset T_f$$ ($$T_\alpha$$ generally exists) will have sets that are open, closed, open and closed, and neither.

What you are describing is correct. A topology is a subset of the power set of a set $$X$$ that has the properties that it contains the empty set and full set, it is closed under arbitrary union of its elements, and it is closed under finite intersection of its elements. An open set is merely an element of a topology.

As you point out, many subsets of the power set may have all of these properties. It is a common early topology problem to find how many topologies can exist on the set $$X = \{1,2,3\}$$.

Since there are different topologies on a set, if you talk about an open subset, you have to make sure you are clear which topology is on the set. Usually, this is clear from context.