public TopologyState GetState()

public void ValidateTopology()
{
  using (Geodatabase geodatabase = new Geodatabase(new FileGeodatabaseConnectionPath(new Uri(@"C:\TestData\GrandTeton.gdb"))))
  using (Topology topology = geodatabase.OpenDataset<Topology>("Backcountry_Topology"))
  {
    // If the topology currently does not have dirty areas, calling Validate() returns an empty envelope.

    ValidationResult result = topology.Validate(new ValidationDescription(topology.GetExtent()));
    Console.WriteLine($"'AffectedArea' after validating a topology that has not been edited => {result.AffectedArea.ToJson()}");

    // Now create a feature that purposely violates the "PointProperlyInsideArea" topology rule.  This action will
    // create dirty areas.

    Feature newFeature = null;

    try
    {
      // Fetch the feature in the Campsites feature class whose objectID is 2.  Then create a new geometry slightly
      // altered from this and use it to create a new feature.

      using (Feature featureViaCampsites2 = GetFeature(geodatabase, "Campsites", 2))
      {
        Geometry currentGeometry = featureViaCampsites2.GetShape();
        Geometry newGeometry = GeometryEngine.Instance.Move(currentGeometry, (currentGeometry.Extent.XMax / 8),
          (currentGeometry.Extent.YMax / 8));

        using (FeatureClass campsitesFeatureClass = featureViaCampsites2.GetTable())
        using (FeatureClassDefinition definition = campsitesFeatureClass.GetDefinition())
        using (RowBuffer rowBuffer = campsitesFeatureClass.CreateRowBuffer())
        {
          rowBuffer[definition.GetShapeField()] = newGeometry;

          geodatabase.ApplyEdits(() =>
          {
            newFeature = campsitesFeatureClass.CreateRow(rowBuffer);
          });
        }
      }

      // After creating a new feature in the 'Campsites' participating feature class, the topology's state should be 
      // "Unanalyzed" because it has not been validated.

      Console.WriteLine($"The topology state after an edit has been applied => {topology.GetState()}");

      // Now validate the topology.  The result envelope corresponds to the dirty areas.

      result = topology.Validate(new ValidationDescription(topology.GetExtent()));
      Console.WriteLine($"'AffectedArea' after validating a topology that has just been edited => {result.AffectedArea.ToJson()}");

      // After Validate(), the topology's state should be "AnalyzedWithErrors" because the topology currently has errors.

      Console.WriteLine($"The topology state after validate topology => {topology.GetState()}");

      // If there are no dirty areas, the result envelope should be empty.

      result = topology.Validate(new ValidationDescription(topology.GetExtent()));
      Console.WriteLine($"'AffectedArea' after validating a topology that has just been validated => {result.AffectedArea.ToJson()}");
    }
    finally
    {
      if (newFeature != null)
      {
        geodatabase.ApplyEdits(() =>
        {
          newFeature.Delete();
        });

        newFeature.Dispose();
      }
    }

    // Validate again after deleting the newly-created feature.

    topology.Validate(new ValidationDescription(topology.GetExtent()));
  }
}

private Feature GetFeature(Geodatabase geodatabase, string featureClassName, long objectID)
{
  using (FeatureClass featureClass = geodatabase.OpenDataset<FeatureClass>(featureClassName))
  {
    QueryFilter queryFilter = new QueryFilter()
    {
      ObjectIDs = new List<long>() { objectID }
    };

    using (RowCursor cursor = featureClass.Search(queryFilter))
    {
      System.Diagnostics.Debug.Assert(cursor.MoveNext());
      return (Feature)cursor.Current;
    }
  }
}