When should I use the HashSet<T> type?

The important thing about HashSet<T> is right there in the name: it's a set. The only things you can do with a single set is to establish what its members are, and to check whether an item is a member.

Asking if you can retrieve a single element (e.g. set[45]) is misunderstanding the concept of the set. There's no such thing as the 45th element of a set. Items in a set have no ordering. The sets {1, 2, 3} and {2, 3, 1} are identical in every respect because they have the same membership, and membership is all that matters.

It's somewhat dangerous to iterate over a HashSet<T> because doing so imposes an order on the items in the set. That order is not really a property of the set. You should not rely on it. If ordering of the items in a collection is important to you, that collection isn't a set.

Sets are really limited and with unique members. On the other hand, they're really fast.

Here's a real example of where I use a HashSet<string>:

Part of my syntax highlighter for UnrealScript files is a new feature that highlights Doxygen-style comments. I need to be able to tell if a @ or \ command is valid to determine whether to show it in gray (valid) or red (invalid). I have a HashSet<string> of all the valid commands, so whenever I hit a @xxx token in the lexer, I use validCommands.Contains(tokenText) as my O(1) validity check. I really don't care about anything except existence of the command in the set of valid commands. Lets look at the alternatives I faced:

• Dictionary<string, ?>: What type do I use for the value? The value is meaningless since I'm just going to use ContainsKey. Note: Before .NET 3.0 this was the only choice for O(1) lookups - HashSet<T> was added for 3.0 and extended to implement ISet<T> for 4.0.
• List<string>: If I keep the list sorted, I can use BinarySearch, which is O(log n) (didn't see this fact mentioned above). However, since my list of valid commands is a fixed list that never changes, this will never be more appropriate than simply...
• string[]: Again, Array.BinarySearch gives O(log n) performance. If the list is short, this could be the best performing option. It always has less space overhead than HashSet, Dictionary, or List. Even with BinarySearch, it's not faster for large sets, but for small sets it'd be worth experimenting. Mine has several hundred items though, so I passed on this.

A HashSet<T> implements the ICollection<T> interface:

public interface ICollection<T> : IEnumerable<T>, IEnumerable
{
    // Methods
    void Add(T item);
    void Clear();
    bool Contains(T item);
    void CopyTo(T[] array, int arrayIndex);
    bool Remove(T item);

    // Properties
   int Count { get; }
   bool IsReadOnly { get; }
}

A List<T> implements IList<T>, which extends the ICollection<T>

public interface IList<T> : ICollection<T>
{
    // Methods
    int IndexOf(T item);
    void Insert(int index, T item);
    void RemoveAt(int index);

    // Properties
    T this[int index] { get; set; }
}

A HashSet has set semantics, implemented via a hashtable internally:

A set is a collection that contains no duplicate elements, and whose elements are in no particular order.

What does the HashSet gain, if it loses index/position/list behavior?

Adding and retrieving items from the HashSet is always by the object itself, not via an indexer, and close to an O(1) operation (List is O(1) add, O(1) retrieve by index, O(n) find/remove).

A HashSet's behavior could be compared to using a Dictionary<TKey,TValue> by only adding/removing keys as values, and ignoring dictionary values themselves. You would expect keys in a dictionary not to have duplicate values, and that's the point of the "Set" part.