diff --git a/README.md b/README.md
index f66fadd..34ee48b 100644
--- a/README.md
+++ b/README.md
@@ -65,8 +65,8 @@ operators like sets do.
     OrderedSet(['r', 'x', 's', 'h', 'z', 'm'])
 
 The `__getitem__()` and `index()` methods have been extended to accept any
-iterable except a string, returning a list, to perform NumPy-like "fancy
-indexing".
+iterable, but not hashable (string, tuple, frozenset) instances, returning a
+list, to perform NumPy-like "fancy indexing".
 
     >>> letters = OrderedSet('abracadabra')
 
diff --git a/ordered_set/__init__.py b/ordered_set/__init__.py
index ccd1cbf..adb6ece 100644
--- a/ordered_set/__init__.py
+++ b/ordered_set/__init__.py
@@ -9,6 +9,7 @@
 from typing import (
     Any,
     Dict,
+    Hashable,
     Iterable,
     Iterator,
     List,
@@ -33,26 +34,8 @@
 OrderedSetInitializer = Union[AbstractSet[T], Sequence[T], Iterable[T]]
 
 
-def _is_atomic(obj: object) -> bool:
-    """
-    Returns True for objects which are iterable but should not be iterated in
-    the context of indexing an OrderedSet.
-
-    When we index by an iterable, usually that means we're being asked to look
-    up a list of things.
-
-    However, in the case of the .index() method, we shouldn't handle strings
-    and tuples like other iterables. They're not sequences of things to look
-    up, they're the single, atomic thing we're trying to find.
-
-    As an example, oset.index('hello') should give the index of 'hello' in an
-    OrderedSet of strings. It shouldn't give the indexes of each individual
-    character.
-    """
-    return isinstance(obj, (str, tuple))
-
-
 class OrderedSet(MutableSet[T], Sequence[T]):
+    __slots__ = ("items", "map")
     """
     An OrderedSet is a custom MutableSet that remembers its order, so that
     every entry has an index that can be looked up.
@@ -232,7 +215,7 @@ def index(self, key):
         Get the index of a given entry, raising an IndexError if it's not
         present.
 
-        `key` can be an iterable of entries that is not a string, in which case
+        `key` can be an iterable of entries that is not a hashable (string, tuple, frozenset), in which case
         this returns a list of indices.
 
         Example:
@@ -240,7 +223,7 @@ def index(self, key):
             >>> oset.index(2)
             1
         """
-        if isinstance(key, Iterable) and not _is_atomic(key):
+        if isinstance(key, Iterable) and not isinstance(key, Hashable):
             return [self.index(subkey) for subkey in key]
         return self.map[key]
 
diff --git a/test/test_ordered_set.py b/test/test_ordered_set.py
index 6efe0a9..eb75a7b 100644
--- a/test/test_ordered_set.py
+++ b/test/test_ordered_set.py
@@ -54,6 +54,14 @@ def test_indexing():
     with pytest.raises(KeyError):
         set1.index("br")
 
+    set2 = OrderedSet((("a", "b"), frozenset(("c", "d")), "efg"))
+    assert set2.index(("a", "b"))==0
+    assert set2.index(frozenset(("c", "d")))==1
+    assert set2.index("efg")==2
+    assert set2.index([frozenset(("c", "d")), ("a", "b")])==[1, 0]
+    assert set2.index(OrderedSet([frozenset(("c", "d")), ("a", "b")]))==[1, 0]
+    with pytest.raises(KeyError):
+        set2.index(["a", "b"])
 
 class FancyIndexTester:
     """