In the new Swift 3 there is now a different initializer for NSNotification (now simply Notification; the NS prefix was dropped):

struct Config {
    static let ShouldCloseBrowserNotification = "ShouldCloseBrowserNotification"
}

let notification = Notification(
    name: Notification.Name(rawValue: Config.ShouldUpdateDialogNotification),
    object: nil,
    userInfo: nil
)
NotificationCenter.default.post(notification)

The name parameter does not look very nice now. But you can extend Notification.Name to make it look better:

extension Notification.Name {
    static let shouldCloseBrowserNotification = Notification.Name("ShouldCloseBrowserNotification")
}

let notification = Notification(
    name: .ShouldUpdateDialogNotification,
    object: nil,
    userInfo: nil
)
NotificationCenter.default.post(notification)

UIKit itself is structured in roughly the same way:

UIKit code