diff --git a/stdlib/2and3/asyncore.pyi b/stdlib/2and3/asyncore.pyi
index 983db7dcf..a5126ccb3 100644
--- a/stdlib/2and3/asyncore.pyi
+++ b/stdlib/2and3/asyncore.pyi
@@ -2,10 +2,11 @@ from typing import Tuple, Union, Optional, Any, Dict, overload
 
 import os
 import select
-import socket
 import sys
 import time
 import warnings
+from socket import SocketType
+from typing import Optional
 
 from errno import (EALREADY, EINPROGRESS, EWOULDBLOCK, ECONNRESET, EINVAL,
                    ENOTCONN, ESHUTDOWN, EINTR, EISCONN, EBADF, ECONNABORTED,
@@ -38,19 +39,20 @@ class dispatcher:
     connecting = ...  # type: bool
     closing = ...  # type: bool
     ignore_log_types = ...  # type: frozenset[str]
+    socket = ...  # type: Optional[SocketType] # undocumented
 
-    def __init__(self, sock: Optional[socket.socket] = ..., map: _maptype = ...) -> None: ...
+    def __init__(self, sock: Optional[SocketType] = ..., map: _maptype = ...) -> None: ...
     def add_channel(self, map: _maptype = ...) -> None: ...
     def del_channel(self, map: _maptype = ...) -> None: ...
     def create_socket(self, family: int, type: int) -> None: ...
-    def set_socket(self, sock: socket.socket, map: _maptype = ...) -> None: ...
+    def set_socket(self, sock: SocketType, map: _maptype = ...) -> None: ...
     def set_reuse_addr(self) -> None: ...
     def readable(self) -> bool: ...
     def writable(self) -> bool: ...
     def listen(self, backlog: int) -> None: ...
     def bind(self, address: Union[tuple, str]) -> None: ...
     def connect(self, address: Union[tuple, str]) -> None: ...
-    def accept(self) -> Optional[Tuple[socket.socket, Any]]: ...
+    def accept(self) -> Optional[Tuple[SocketType, Any]]: ...
     def send(self, data: bytes) -> int: ...
     def recv(self, buffer_size: int) -> bytes: ...
     def close(self) -> None: ...
@@ -108,7 +110,7 @@ class dispatcher:
         def shutdown(self, how: int) -> None: ...
 
 class dispatcher_with_send(dispatcher):
-    def __init__(self, sock: socket.socket = ..., map: _maptype = ...) -> None: ...
+    def __init__(self, sock: SocketType = ..., map: _maptype = ...) -> None: ...
     def initiate_send(self) -> None: ...
     def handle_write(self) -> None: ...
     # incompatible signature: