diff --git a/fsspec/caching.py b/fsspec/caching.py
index 2812724fc..8d441fe46 100644
--- a/fsspec/caching.py
+++ b/fsspec/caching.py
@@ -111,7 +111,7 @@ def _makefile(self) -> mmap.mmap | bytearray:
             fd.write(b"1")
             fd.flush()
         else:
-            fd = open(self.location, "rb+")
+            fd = open(self.location, "r+b")
 
         return mmap.mmap(fd.fileno(), self.size)
 
diff --git a/fsspec/implementations/cached.py b/fsspec/implementations/cached.py
index 89fab0791..eef916bb2 100644
--- a/fsspec/implementations/cached.py
+++ b/fsspec/implementations/cached.py
@@ -771,10 +771,10 @@ def __init__(self, fs, path, fn=None, mode="wb", autocommit=True, seek=0):
         self.autocommit = autocommit
 
     def __reduce__(self):
-        # always open in rb+ to allow continuing writing at a location
+        # always open in r+b to allow continuing writing at a location
         return (
             LocalTempFile,
-            (self.fs, self.path, self.fn, "rb+", self.autocommit, self.tell()),
+            (self.fs, self.path, self.fn, "r+b", self.autocommit, self.tell()),
         )
 
     def __enter__(self):
diff --git a/fsspec/implementations/http.py b/fsspec/implementations/http.py
index cdd84c5ce..7b249bd94 100644
--- a/fsspec/implementations/http.py
+++ b/fsspec/implementations/http.py
@@ -802,7 +802,7 @@ async def get_range(session, url, start, end, file=None, **kwargs):
     async with r:
         out = await r.read()
     if file:
-        with open(file, "rb+") as f:
+        with open(file, "r+b") as f:
             f.seek(start)
             f.write(out)
     else:
diff --git a/fsspec/implementations/memory.py b/fsspec/implementations/memory.py
index 32daaf0f1..4022ff13a 100644
--- a/fsspec/implementations/memory.py
+++ b/fsspec/implementations/memory.py
@@ -175,7 +175,7 @@ def _open(
             parent = self._parent(parent)
             if self.isfile(parent):
                 raise FileExistsError(parent)
-        if mode in ["rb", "ab", "rb+"]:
+        if mode in ["rb", "ab", "r+b"]:
             if path in self.store:
                 f = self.store[path]
                 if mode == "ab":
diff --git a/fsspec/implementations/tests/test_memory.py b/fsspec/implementations/tests/test_memory.py
index 05a40b287..038d72eaa 100644
--- a/fsspec/implementations/tests/test_memory.py
+++ b/fsspec/implementations/tests/test_memory.py
@@ -191,6 +191,15 @@ def test_seekable(m):
     assert f.tell() == 2
 
 
+# https://github.com/fsspec/filesystem_spec/issues/1425
+@pytest.mark.parametrize("mode", ["r", "rb", "w", "wb", "ab", "r+b"])
+def test_open_mode(m, mode):
+    filename = "mode.txt"
+    m.touch(filename)
+    with m.open(filename, mode=mode) as _:
+        pass
+
+
 def test_remove_all(m):
     m.touch("afile")
     m.rm("/", recursive=True)