f is one-to-one by the definition of a function, and it’s obviously onto. We just have to show f and f −1 |Γf (E1 ) are continuous. The finite open intervals (a, b) form a basis of E 1 . Let Ua,b = n (x, y) ∈ E 2 | x ∈ (a, b) o . Then U is open in E 2 . Sets of the form B∩Γf(E 1 ) form a base of open sets in Γf(E 1 ), where B is an open ball in E 2 . Note that B∩Γf(E 1 ) = Ua,b ∩Γf(E 1 ) for some Ua,b. And then Γ −1 f (B ∩ Γf(E 1 )) = Γ−1 f (Ua,b) = (a, b) is open in E 1 . Thus Γf is continuous. Conversely, f((a, b)) = Γf(E 1 ) ∩ Ua,b. Therefore f((a, b)) is open in Γf(E 1 ). Thus Γ −1 f is continuous. Thus Γf is a homeomorphism.

这是一个数学问题，我可以回答。根据定义，如果一个函数 f 是一对一的，且显然是到达的，我们只需要证明 f 和 f-1 |Γf(E1) 是连续的。有限开区间 (a, b) 形成 E1 的基础。让 Ua，b = n (x，y) ∈ E2 | x ∈ (a，b) o。然后 U 在 E2 中是开放的。形式为 B∩Γf(E1) 的集合形成 Γf(E1) 中开放集的基础，其中 B 是 E2 中的开放球。请注意，B∩Γf(E1) = Ua，b ∩Γf(E1) 对于某些 Ua，b。然后 Γ−1f(B∩Γf(E1)) = Γ−1f(Ua，b) = (a，b) 在 E1 中是开放的。因此 Γf 是连续的。反之，f((a，b)) = Γf(E1) ∩ Ua，b。因此 f((a，b)) 在 Γf(E1) 中是开放的。因此 Γ−1f 是连续的。因此 Γf 是一个同胚。

相关问题

GM-GRA-DPC-PSOSVR forecasting model For any regional carbon emission forecasting, it is important to select a suitable model. For model selection, researchers often choose only one optimal model according to experience or model selection criteria, which obviously ignores the uncertainty of model selection. That is, once the data changes slightly, the optimal model may change, which means the optimal model is uncertain. The combined forecasting was proposed by Bates and Granger(1969), which is a robust forecasting technique.

这段话也没有发现任何语法错误。该段介绍了GM-GRA-DPC-PSOSVR预测模型的构建。对于任何地区的碳排放预测，选择一个合适的模型是非常重要的。在模型选择方面，研究人员通常根据经验或模型选择标准选择一个最优模型，这显然忽略了模型选择的不确定性。也就是说，一旦数据发生细微变化，最优模型可能会发生变化，这意味着最优模型是不确定的。Bates和Granger(1969)提出了组合预测方法，这是一种强大的预测技术。

Also create a ControllerMoreBook class that extends Controller.The moreBook method takes the name of a user and a number of books (as a string) as arguments. The moreBook method of the controller then transforms the number of books from a string to an integer (using the Integer.parseInt static method) and calls the moreBook method of the library to increase the number of books borrowed or lent by the user (depending on what kind of user it is) of a specific user, by the given argument. • If no exception occurs then the moreBook method of the controller returns the empty string. • If the moreBook method of the library throws an UnknownUserException then the moreBook method of the controller must catch this exception and return as result the error message from the exception object. • If the moreBook method of the library throws a NotALenderException then the moreBook method of the controller must catch this exception and return as result the error message from the exception object. • If the parseInt method of the Integer class throws a NumberFormatException (because the user typed something which is not an integer) then the moreBook method of the controller must catch this exception and return as result the error message from the exception object. Note: to keep things simple, it is allowed for a user of your system to increase the number of books of a user by a negative number, so there is no need to check for that. Modify the run method of the GUI class to add a ViewMoreBook view that uses a ControllerMoreBook controller and the same model as before (not a new model!) Do not delete the previous views. Run your GUI and check that you can correctly use the new view to increase the number of books for different users of your library (obviously your library must have some users in it to test this: see the last paragraph of Question 7). • Check that, when you increase a user’s book, the simple view is automatically correctly updated to show the new total number of borrowed books for all users of the library. • Also use the “get book” view to check that the user’s book value correctly changed. • Also check that increasing the book number of an unknown user correctly shows an error message.Also check that increasing the book of a user by a large negative number correctly shows an error message. Also check that trying to increase the book of a user by a number which is not an integer correctly shows an error message (do not worry about the content of the error message). 完成符合以上要求的java代码

Library.java:

import java.util.HashMap;
import java.util.Map;

public class Library {
    private Map<String, User> users;

    public Library() {
        users = new HashMap<>();
    }

    public void addUser(User user) {
        users.put(user.getName(), user);
    }

    public void moreBook(String name, int num) throws UnknownUserException, NotALenderException {
        User user = users.get(name);
        if (user == null) {
            throw new UnknownUserException(name);
        }
        if (!(user instanceof Lender)) {
            throw new NotALenderException(name);
        }
        Lender lender = (Lender) user;
        lender.moreBook(num);
    }

    public int getNumBorrowed(String name) throws UnknownUserException {
        User user = users.get(name);
        if (user == null) {
            throw new UnknownUserException(name);
        }
        return user.getNumBorrowed();
    }
}

User.java:

public abstract class User {
    private String name;
    private int numBorrowed;

    public User(String name) {
        this.name = name;
        numBorrowed = 0;
    }

    public String getName() {
        return name;
    }

    public int getNumBorrowed() {
        return numBorrowed;
    }

    public void setNumBorrowed(int numBorrowed) {
        this.numBorrowed = numBorrowed;
    }
}

Lender.java:

public class Lender extends User {
    public Lender(String name) {
        super(name);
    }

    public void moreBook(int num) {
        setNumBorrowed(getNumBorrowed() + num);
    }
}

UnknownUserException.java:

public class UnknownUserException extends Exception {
    private String name;

    public UnknownUserException(String name) {
        this.name = name;
    }

    public String getMessage() {
        return "Unknown user: " + name;
    }
}

NotALenderException.java:

public class NotALenderException extends Exception {
    private String name;

    public NotALenderException(String name) {
        this.name = name;
    }

    public String getMessage() {
        return name + " is not a lender";
    }
}

ControllerMoreBook.java:

public class ControllerMoreBook extends Controller {
    private Library library;

    public ControllerMoreBook(Library library) {
        this.library = library;
    }

    public String moreBook(String name, String numStr) {
        try {
            int num = Integer.parseInt(numStr);
            library.moreBook(name, num);
            return "";
        } catch (UnknownUserException e) {
            return e.getMessage();
        } catch (NotALenderException e) {
            return e.getMessage();
        } catch (NumberFormatException e) {
            return "Invalid number format";
        }
    }
}

ViewMoreBook.java:

import javax.swing.*;
import java.awt.*;

public class ViewMoreBook extends JFrame implements View {
    private JTextField nameField;
    private JTextField numField;
    private JButton okButton;
    private JTextArea resultArea;
    private ControllerMoreBook controller;

    public ViewMoreBook(ControllerMoreBook controller) {
        this.controller = controller;

        setTitle("More Books");
        setSize(300, 200);
        setDefaultCloseOperation(JFrame.EXIT_ON_CLOSE);

        JPanel panel = new JPanel();
        panel.setLayout(new GridLayout(3, 2));
        panel.add(new JLabel("Name:"));
        nameField = new JTextField();
        panel.add(nameField);
        panel.add(new JLabel("Number of Books:"));
        numField = new JTextField();
        panel.add(numField);
        okButton = new JButton("OK");
        panel.add(okButton);
        resultArea = new JTextArea();
        resultArea.setEditable(false);
        panel.add(resultArea);

        add(panel);

        okButton.addActionListener(e -> {
            String name = nameField.getText();
            String numStr = numField.getText();
            String result = controller.moreBook(name, numStr);
            resultArea.setText(result);
        });
    }
}

修改 GUI 类的 run 方法：

public void run() {
    Library library = new Library();
    library.addUser(new Lender("Alice"));
    library.addUser(new Lender("Bob"));
    library.addUser(new Borrower("Charlie"));

    ControllerSimple simpleController = new ControllerSimple(library);
    ViewSimple simpleView = new ViewSimple(simpleController);
    simpleView.setVisible(true);

    ControllerGetBook getBookController = new ControllerGetBook(library);
    ViewGetBook getBookView = new ViewGetBook(getBookController);
    getBookView.setVisible(true);

    ControllerMoreBook moreBookController = new ControllerMoreBook(library);
    ViewMoreBook moreBookView = new ViewMoreBook(moreBookController);
    moreBookView.setVisible(true);
}