The extern variable causes a multiple definition error

I am trying to use extern to use a variable that was previously defined.

I have not used extern before and now I need to use it to define a variable only once and use them in several files

I wrote a minimal version of the code for this question. I have four files

lib.h

#ifndef LIB_H
#define LIB_H

#include <iostream>

namespace lib {

  extern bool initialized;

  bool initialized = false;

  static void isInit(char* parent) {
    std::cout << "Library for [" << parent << "] initialized? " << (::lib::initialized ? "yes" : "no") << "\n";
  }
} // namespace lib
#endif

vehicle.h

#ifndef _VEHICLE_H
#define _VEHICLE_H
#include <string>

class Vehicle {
  public:
    Vehicle(const std::string& manufacturer,
            const std::string& model,
            int year);
    std::string manufacturer;
    std::string model;
    int year; 
};
#endif

Below is the version of vehicle.h called vehicle.cpp

#include "vehicle.h"

#include "lib.h"

Vehicle::Vehicle(const std::string& manufacturer,
                 const std::string& model,
                 int year) :
                    manufacturer(manufacturer),
                    model(model),
                    year(year) {
   ::lib::isInit("Vehicle");
}

main.cpp

#include "vehicle.h"

#include "lib.h"

int main(int argc, char** argv) {

   ::lib::isInit("main");

   ::lib::initialized = true;

   ::lib::isInit("main");

   Vehicle vehicle("Toyota", "Corolla", 2013);

   return 0;
}

I am using g ++

g++ -Wno-write-strings main.cpp vehicle.cpp -o bin/main.cpp.bin 

I get the following errors:

/tmp/cclVpsgT.o:(.bss+0x0): multiple definition of `lib::initialized'
/tmp/ccmJKImL.o:(.bss+0x0): first defined here
collect2: error: ld returned 1 exit status

I checked the output:

g++ -Wno-write-strings main.cpp vehicle.cpp -E

multiple definitions occur every time lib.h is enabled.

My questions:

• Why is lib.h turned on several times when the guard parameter exists.
• How would I define an extern variable and initialize it in the same file (since it was used in the same file later)