# Terminology when talking about topological spaces

I am reading through Munkres: Topology a First Course, and I find the convention of shortening sentences to represent extremely frustrating. I will give an example below

When dealing with a topological space $X$, and a subspace $Y$, one needs to exercise care in taking closures of sets. If $A$ is a subset of $Y$....

Now the last part can mean two very different things, and I will translate the rest of this sentence to show that.

My translation of the above sentence: When dealing with a topological space $(X, \mathcal{T_x})$, and a subspace $(Y \subset X, \mathcal{T_y})$ where $\mathcal{T_y} = \{Y \cap U \ | \ U \in \mathcal{T_x}\}$, one needs to exercise care in taking closures of sets.

Now in my translation of what Munkres has written, "If $A$ is a subset of $Y$" could mean one of two things, either

1. $A \subset Y$
2. $A \subset \mathcal{T_y}$

$(1)$ Above is a subset of $Y$, the underlying set of the subspace, whereas $(2)$ above is a subset of the topology $\mathcal{T_y}$ of the subspace and is a collection of subsets of $Y$, very different to the meaning $(1)$ has.

But we could be truly pedantic and write the topological space $(X, \mathcal{T_x})$ as $$(X, \mathcal{T_x}) = \{ \{X\}, \{X, \mathcal{T}\}\}$$, by the set-theoretic definition of an ordered-pair, and then we would have only have one of four possibilities for for a subset of the topological space

1. $\emptyset \subset \{ \{X\}, \{X, \mathcal{T}\}\}$
2. $\{\{X\}\} \subset \{ \{X\}, \{X, \mathcal{T}\}\}$
3. $\{\{X, \mathcal{T}\}\} \subset \{ \{X\}, \{X, \mathcal{T}\}\}$
4. $\{ \{X\}, \{X, \mathcal{T}\}\} \subset \{ \{X\}, \{X, \mathcal{T}\}\}$

In this pedantic case, the words subset of a topological space, doesn't have the familiar meaning that we assosciate with it.

Now the sad part is that this convention/terminology (the original one quoted right at the top) is used by most mathematicians, and I feel it is something that I'm just going to have to get used to.

So my question boils down to this. I'll use the convention of a topological space being defined as an ordered pair $(X, \mathcal{T})$ for the purposes and context of this question. So when someone says subset of a topological space $X$, or subset of a subspace $Y$, are they

1. Referring to the underlying set $X$ in the topological space?
2. Or are they referring to the topology $\mathcal{T}$ on the set in the topological space?
• The statement "$A$ is a subset of $Y$" seem very clear to me. Why you can think that $A\subset\mathcal T_y$ instead of $A\subset Y$? – Masacroso Jan 2 '17 at 11:31
• $A$ is a subset of $Y$ cannot possibly mean $A\subseteq\mathcal{T}_y$; it can only mean exactly what it says, $A\subseteq Y$. – Brian M. Scott Jan 2 '17 at 12:00
• A generic subset of $Y$ corresponds to the formulation $A\subseteq Y$. A generic open subset of $Y$ corresponds to the formulation $A\in{\cal T}_Y$, while $A\subseteq{\cal T}_Y$ would specify a certain collection of open subsets in $Y$. – Andrea Mori Jan 2 '17 at 12:28
• Did you ever wonder what a subset of a group is? – Carsten S Jan 2 '17 at 14:59

Mathematical language and the understanding thereof, like a lot of human language, depends on context. The eyes read something, the brain processes, and one learns some mathematics. You're right that you'll just have to get used to it, just like you've gotten used to the contexts and conventions of human language in other, non-mathematical settings.

Here's an example of how that works for me, with the text that you have provided, but without any symbols other than the ones provided in the text.

Step 1: My eyes read: "When dealing with a topological space $X$..."

Step 2: My brain processes: $X$ is a set with a topology. That topology, while un-named, is given, i.e. it is uniquely associated to $X$. Although that topology is not named, in any future context where $X$ is discussed, the topology associated to $X$ will be that uniquely associated, given topology.

Step 3: My eyes continue to read: "...and a subspace $Y$..."

Step 4: My brain processes: $Y$ is a subset of $X$, $Y$ is also a topological space, and the topology on $Y$ is the subspace topology induced by the given topology on $X$ (see Step 2). Although that topology is not named, in any future context where $Y$ is discussed, the topology associated to $Y$ will be the subspace topology induced by the given topology on $X$.

Step 5: My eyes continue to read: "...one needs to exercise care in taking closure of sets. If $A$ is a subset of $Y$..".

Step 6: My brain processes: Closure of a subset requires that the subset be given as a subset of a particular topological space. In the present context there are two topological spaces under discussion, namely the topological space $X$ and its subspace $Y$. The set $A$ is both a subset of both $Y$ and $X$ --- in fact, every subset of $Y$ is a subset of $X$. So I am being warned about the fact that the closure of $A$ could mean closure with respect to the given topology on $X$ or closure with respect to the subspace topology on $Y$. These two closures of $A$ could be different. So in this context I am being warned about future contexts where there could be an ambiguity in the usage of the term "closure".

Summary: What the context provides in this example is a particular topology associated to both $X$ and $Y$. My brain knows this and sees no ambiguities --- despite the lack of symbols on the page to represent those topologies --- and in any future discussions of $X$ and $Y$ my brain keeps track of the topology on $X$ and the topology on $Y$.