I wrote a Processing sketch to help demo and build intuition about how topological data analysis works (in two dimensions). It builds on the javaplexDemo.pde written by Mikael Vejdemo-Johansson. You can add and remove data points and see the persistent homology barcode generated and plotted in real-time. You can save datasets and barcodes, to compare them. It provides a nice way to visualize the filtration parameter.
To see screenshots and a screencast, check out this page. The GitHub repo for the project is here.