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Right now I'm getting into topology since I want to understand it more prior to fully learning Real Analysis.

I'm confused about how to construct both a Sub-Basis and a Base/Basis for a topology $X$.

I read the definition for Basis and Sub-Basis and find it a little confusing.

For example:

Definition. A subbasis $S$ for a topology on $X$ is a collection of subsets of $X$ whose union equals $X$. The topology generated by the subbasis $S$ is defined to be the collection $T$ of all unions of finite intersections of elements of $S$.

So with this definition I'm confused about the second part where it describes the unions of finite intersections of elements of $S$. And what I'm grasping from the first part is that a subbasis contains elements that are in $X$ but there union creates the set $X$.

so If I have the set $X = \{a,b,c,d\}$ then $T = \{X,\varnothing, \{c,d\},\{b\}\}$ is a topology on $X$.

Then the subbasis $S = \{\{a\},\{b\},\{c\},\{d\}\}$.

I'm not sure if this is correct.

EDIT Suppose $X = \{a,b,c,d,e\}$ then $B = \{\{a\},\{a,c\},\{a,b\}$,$\{b\},\{d\}$,$\{e\},${c}$\}.$ is a basis for X.

Is this a basis or simply nothing?

I'm currently using James Munkres Topology, 2nd Economy Edition.

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    $\begingroup$ Note that when you take the set of union of intersections of elements in your subbasis, the set should be equal to your topology. Also the topology you gave isn't a topology because it is not closed under union. $\endgroup$ – Enigma Jan 10 '15 at 19:14
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I think you are confused about the distinctions between a set $X$, a topology on $X$, and a topological space.

The reason I think you’re confused is that you make several statements that don’t make sense:

  • “a topology $X$” (instead of a topology on $X$ or for $X$).
  • “a subbasis contains elements that are in X but there [sic] union creates the set X” A subbasis for a topology on $X$ is a collection of subsets of $X$, not elements of $X$.*
  • “$B$ is a basis for $X$.” Where you say this, you haven’t mentioned any topology for $X$. The assertion that $B$ is a basis for $X$ doesn’t exactly make sense on its own. You can say that $B$ is a basis for a topological space $(X,T)$, and this could be abbreviated to saying $B$ is a basis for $X$, but only if there is a clear context where $X$ is a topological space with some topology $T$.

A topology for $X$ is a collection of subsets of $X$ that includes the sets $\emptyset$ and $X$ among its elements and that is closed under arbitrary unions and finite intersections.

A topological space is a set $X$ together with a topology $T$ for $X$. While it’s best to say “the topological space $(X,T)$,” mathematicians do say “the topological space $X$” without mentioning $T$ if it’s clear what $T$ is.

Be sure you understand the concept of a topology first.

*Incidentally, the word “contains” can be unclear in mathematics, as it can mean “contains as elements” or “contains as a subset.” For example, $\mathbb{R}$ contains (as a subset) $\mathbb{Q}$ and contains (as an element) $\frac{1}{2}$ and every other rational number.

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I think it will help to understand exactly what your after when considering a subbase. Start with choosing a random collection of subsets $S$ of $X$. In general, you know that this collection $S$ does not form a topology. Now comes the fundamental question:

"What do we have to change about $S$ to get a topology?"

Well, for one, arbitrary unions and finite intersections of elements of $S$ ought to again be in $S$. If they are not, we simply add them to $S$ to get a new collection $T$.

Now we also need $X$ and the empty set $\emptyset$ to be in $T$ as well. However, if we add the condition that the union of elements of $S$ is $X$, we know that $X$ will be in $T$ by construction. This is what motivates the definition of a subbase.

In your example, $T=\{X,\emptyset, \{c,d \}, \{b\}\}$ is not a topology since the union $$\{c,d\}\cup \{b\}=\{c,b,d\}\not \in T.$$ Instead, let's consider $$T=\{\emptyset, \{b\}, \{c,d \}, \{c,b,d\}, X\}.$$ Does this form a topology? Check the axioms. Is your original example $$S=\{X,\emptyset, \{c,d \}, \{b\}\}$$ a subbase for T?

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