The snow may have been flying in upstate New York on Monday, but the mud was still being slung and was very visible in the special election to replace now-Sen. Kirsten Gillibrand (D) in the House.

With the election just four weeks away, there appear to be significant developments every few hours.

Among the most recent:

• Former New York Mayor Rudy Giuliani (R) braved the snow to stump in Dutchess County on Monday, with state Assembly Minority Leader James Tedisco, the Republican nominee in the special election. The duo appeared together at a business roundtable at a construction company in Pleasant Valley, at a meeting on homeland security issues at a fire station in Poughkeepsie and at a rally at another fire department in Hyde Park.

• Unsurprisingly, the Democratic nominee, venture capitalist Scott Murphy, was endorsed Monday by the New York AFL-CIO, and more significantly, he earned the nomination over the weekend of the centrist New York Independence Party. That gives Murphy a third ballot line in the March 31 election, something Gillibrand did not enjoy during her House victories in 2006 and 2008.

• The Democratic Congressional Campaign Committee over the weekend began airing a 60-second radio spot hitting Tedisco for not saying whether he’d have supported the economic stimulus package that passed Congress in February. The National Republican Congressional Committee went on the air last week with its own radio spot attacking Murphy for his business practices and tax record.

Given the upstate district’s political overlay, with 70,000 more enrolled Republicans than Democrats, Tedisco is favored to replace Gillibrand. But there is plenty going on in the race that makes both parties nervous.