Using transport="stdio" closes real stdio, causing ValueError after server exits

Initial Checks

• I confirm that I'm using the latest version of MCP Python SDK
• I confirm that I searched for my issue in https://github.com/modelcontextprotocol/python-sdk/issues before opening this issue

Description

Hi! I ran into an issue where running the server with transport="stdio" causes subsequent stdio operations to fail after the server exits.

Minimal reproduction

from mcp.server.fastmcp import FastMCP

mcp = FastMCP("Demo")
mcp.run(transport="stdio")
print("?")

When using Ctrl+D to exit the server, the final print raises:

Traceback (most recent call last):
  File "...", line 5, in <module>
    print("?")
ValueError: I/O operation on closed file.

I’m not sure if this is intended behavior though.

Likely cause

In stdio.py, stdio is wrapped like this:
https://github.com/modelcontextprotocol/python-sdk/blob/5301298225968ce0fa8ae62870f950709da14dc6/src/mcp/server/stdio.py#L44-L47

When these wrappers are closed, they also close the sys.stdin.buffer / sys.stdout.buffer.

Proposed fix

    if not stdin:
        stdin_fd = os.dup(sys.stdin.fileno())
        stdin_bin = os.fdopen(stdin_fd, "rb", closefd=True)
        stdin = anyio.wrap_file(TextIOWrapper(stdin_bin, encoding="utf-8"))
    if not stdout:
        stdout_fd = os.dup(sys.stdout.fileno())
        stdout_bin = os.fdopen(stdout_fd, "wb", closefd=True)
        stdout = anyio.wrap_file(TextIOWrapper(stdout_bin, encoding="utf-8"))

Python & MCP Python SDK

Python 3.10.15
mcp==1.25.0

從 src/mcp/server/stdio.py 中連結行附近的 wrapper 設定開始，然後執行 issue 中的最小 FastMCP 重現。確認關閉 stdio wrapper 不再關閉程序的實際標準串流，並驗證伺服器結束後最後的 print 能夠成功執行。