When you create a new ObservableCollection with another collection you are not wrapping that collection, you create a new one where all items of the passed collection are copied to the ObservableCollection. If you want to use an ObservableCollection for the sole purpose of DataBinding, look no further, you can bind to any IEnumerable in WPF. This unfortuantely has the drawback that WPF will not always correctly pickup changes to the bound collection. If this is an issue you d probably have to create your own obeservable hashset:

public class ObservableHashSet<T> : ObservableCollection<T>  
{ 
    protected override void InsertItem(int index, T item) 
    { 
        if (Contains(item)) 
        {
            throw new ItemExistsException(item); 
        }
        base.InsertItem(index, item); 
    } 

    protected override void SetItem(int index, T item) 
    { 
        int i = IndexOf(item); 
        if (i >= 0 && i != index)
        {
             throw new ItemExistsException(item); 
        }       
        base.SetItem(index, item); 
    } 
}

EDIT: AS already has been pointed out, you can not inherit from HashSet to implement INotifyCollectionChanged. However if you look at the code (using Reflector) for the HashSet class it is pretty simple it should be too hard to mimic that functionality yourself.

As per bitbonk s answer, but I wanted to override the add(T item) method, but you can t, so I created an append(T item) method instead:

public class ObservableSetCollection<T> : ObservableCollection<T> {
    public void Append(T item) {
        if (Contains(item)) return;
        base.Add(item);
    }
}

And then in my code behind:

public partial class MainWindow : Window {
    private ObservableSetCollection<string> consolidationHeaders;

    public MainWindow() {
        InitializeComponent();
        initialize();
    }

    private void initialize() {
        consolidationHeaders = new ObservableSetCollection<string>();
        listboxConsolidationColumns.ItemsSource = consolidationHeaders;
    }

    .
    .
    .


    private void listboxAvailableColumns_MouseDoubleClick(object sender, MouseButtonEventArgs e) {
        consolidationHeaders.Append(listboxAvailableColumns.SelectedValue.ToString());
    }

    private void listboxConsolidationColumns_MouseDoubleClick(object sender, MouseButtonEventArgs e) {
        consolidationHeaders.Remove(listboxConsolidationColumns.SelectedValue.ToString());
    }
}

In the above I have two listboxes, listboxAvailableColumns, which has a list of strings that the user can select by double-clicking, which adds the selection to the second listbox, listboxConsolidationColumns. No duplicates are allowed, and this works perfectly with the ObservableSetCollection exactly as above.

The xaml is simply:

<Grid Margin="5,5,5,5">
    <Grid.ColumnDefinitions>
        <ColumnDefinition Width="1*" />
        <ColumnDefinition Width="1*" />
    </Grid.ColumnDefinitions>
    <Grid.RowDefinitions>
        <RowDefinition Height="Auto" />
        <RowDefinition Height="1*" />
    </Grid.RowDefinitions>
    <Label Grid.Row="0" Grid.Column="0" Margin="0,0,0,0" VerticalAlignment="Center" HorizontalAlignment="Center" Content="Available Columns"/>
    <Label Grid.Row="0" Grid.Column="1" Margin="0,0,0,0" VerticalAlignment="Center" HorizontalAlignment="Center" Content="Consolidation Columns"/>
    <ListBox  Grid.Row="1" Grid.Column="0" Name="listboxAvailableColumns" MouseDoubleClick="listboxAvailableColumns_MouseDoubleClick" />
    <ListBox  Grid.Row="1" Grid.Column="1" Name="listboxConsolidationColumns" MouseDoubleClick="listboxConsolidationColumns_MouseDoubleClick" />
</Grid>

As bitbonk said, ObservableCollection doesn t wrap the Hashset but copies its elements instead.

If you want an Observable Hashset check out How can I make an Observable Hashset in C# ?