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This is a way of making new surfaces out of old ones

**Definition**

Given two surfaces *S*_{1} and *S*_{2} the **connected sum** *S*_{1} # *S*_{2} is constructed by removing a disc from each and then joining them along the boundaries of the holes.

**Examples**

- The connected sum of two tori is a 2-holed torus.

- The sphere is an identity for the connected sum operation: [
*S*^{2}- (disc) disc ]

If we start with a projective plane and remove a disc we get a Möbius band.

We can split a Klein bottle into a pair of Möbius bands.Hence a connected sum of Projective planes is a Klein bottle.

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