diff --git a/pymongo/asynchronous/mongo_client.py b/pymongo/asynchronous/mongo_client.py
index 8319755d0f..2f13b92c6b 100644
--- a/pymongo/asynchronous/mongo_client.py
+++ b/pymongo/asynchronous/mongo_client.py
@@ -854,6 +854,7 @@ def __init__(
             server_monitoring_mode=options.server_monitoring_mode,
         )
 
+        self._opened = False
         self._init_background()
 
         if _IS_SYNC and connect:
@@ -1528,9 +1529,11 @@ async def _get_topology(self) -> Topology:
         If this client was created with "connect=False", calling _get_topology
         launches the connection process in the background.
         """
-        await self._topology.open()
-        async with self._lock:
-            self._kill_cursors_executor.open()
+        if not self._opened:
+            await self._topology.open()
+            async with self._lock:
+                self._kill_cursors_executor.open()
+            self._opened = True
         return self._topology
 
     @contextlib.asynccontextmanager
diff --git a/pymongo/synchronous/mongo_client.py b/pymongo/synchronous/mongo_client.py
index 69bfa7a9d5..20d1b3f82f 100644
--- a/pymongo/synchronous/mongo_client.py
+++ b/pymongo/synchronous/mongo_client.py
@@ -853,6 +853,7 @@ def __init__(
             server_monitoring_mode=options.server_monitoring_mode,
         )
 
+        self._opened = False
         self._init_background()
 
         if _IS_SYNC and connect:
@@ -1527,9 +1528,11 @@ def _get_topology(self) -> Topology:
         If this client was created with "connect=False", calling _get_topology
         launches the connection process in the background.
         """
-        self._topology.open()
-        with self._lock:
-            self._kill_cursors_executor.open()
+        if not self._opened:
+            self._topology.open()
+            with self._lock:
+                self._kill_cursors_executor.open()
+            self._opened = True
         return self._topology
 
     @contextlib.contextmanager