Yesterday, I introduced the idea of a *metric space*, and then used it to define *open* and *closed* sets in the space. (And of course, being a bozo, I managed to include a typo that made the definition of open sets equivalent to the definition of closed sets. It’s been corrected, but if you’re not familiar with this stuff, you might want to go back and take a look at the corrected version. It’s just replacing a ≤ with a <, but that makes a *big* difference in meaning!)
\nToday I’m going to explain what a *topological space* is, and what *continuity* means in topology. (For another take on continuity, head on over to [Antopology][antopology] where Ofer has posted his explanation.)
\nA *topological space* is a set **X** and a collection **T** of subsets of **X**, where the following conditions hold:
\n1. ∅ ∈ **T** and **X** ∈ **T*.
\n2. ∀ C ∈ ℘(**T**); : ∪(c ∈ C) ∈ **T**. (That is, the union of any collection of subsets of **T** is an element of **T**. )
\n3. ∀ s, t ∈ **T** : s ∩ t ∈ T. *(The intersection of any two subsets of **T** is also in **T**.)
\nThe collection **T** is called a *topology* on **T**. The *members* of **T** are the *open sets* of the topology. The *closed sets* are the set complements of the members of **T**. Finally, the *elements* of the topological space **X** are called *points*.
\nThe connection to metric spaces should be pretty obvious. The way we built up open and closed sets over a metric space can be used to produce topologies. The properties we worked out for the open and closed sets are exactly the properties that are *required* of the open and closed sets of the topology. There are many ways to build a topology other than starting with a metric space, but that’s definitely the easiest way.
\nOne of the most important ideas in topology is the notion of *continuity*. In some sense, it’s *the* fundamental abstraction of topology. We can now define it.
\nA *function* from topological space **X** to topological space **U** is *continuous* if\/f for every open sets C ∈ **T** the *inverse image* of f on C is an open set. The inverse image is the set of points x in **X** where f(x) ∈ C.
\nThat’s a bit difficult to grasp. What it’s really capturing is that there are no *gaps* in the function. If there were a gap, then the open spaces would no longer be open. Think of the metric spaces idea of open sets. Imagine an open set with a cube cut out of the middle. It’s definitely not continuous. If you took a function on that open set, and its inverse image was the set with the cube cut out, then the function is not smoothly mapping from the open set to the other topological space. It’s mapping *part of* the open set, leaving a big ugly gap.
\nNow, here’s were it gets kind of nifty. The set of of topological spaces and continuous functions form a *category*. with the spaces as objects and the functions as morphisms. We call this category **Top**. It’s often easiest to talk about topological spaces using the constructs of category theory.
\nSo, for example, one of the most important ideas in topology is *homeomorphism*. A homeomorphism is a bicontinuous bijection (a bicontinuous function is a continuous function with a continuous inverse; a bijection is a bidirectional total function between sets.) A homeomorphism between topological spaces is a *homomorphism* in **Top**.
\nFrom the perspective of topology, any two topological spaces with a homeomorphism between them are *identical*. (Which ends up corresponding exactly to how we defined the idea of *equality* in category theory.)
\n[antopology]: http:\/\/antopology.blogspot.com\/2006\/08\/continuity-introduced.html<\/p>\n","protected":false},"excerpt":{"rendered":"
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