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Point set topology

Motivation

Coffee cup and donut are geometrically different, but topologically same: isotopes! Can deform one to the other. Generalize notions of convergence, connectedness, continuity.

The topological space

Set of points or Topological space X. Topology T: Class of some sets of points closed under \(\union, \inters\).

Sets \(S_{i}\) in T are said to be open. \(S_{i}’\) are closed sets. Neighborhood of p is a set \(V \supset\) open set \(U \ni p\). Similarly define nbd of set S of points. \(A \in X\) is dense if any nbd has some \(a\in A\).

Spanning set (of sets); its linear span. Basis of topology.

Topological Morphisms

Every ‘object’ in \(Y\) is a continuous function \(f: \)X\( \to Y\), where \(X\) and \(Y\) are topological spaces. A tea-cup is a function to \(R^{3}\).

Homotopy

Take 2 objects/ cont functions \(f, g: \)X\( \to Y\). Homotopy is continuous function \(H:X \times [0, 1] \to Y\), with \(H(x, 0) = f; H(x, 1) = g\). Think of second parameter as time, and H as a continuous deformation.

If \(H(x,t)\) is also 1:1, \(H\) is an isotopy.

Continuous morphism

f(x) neighborhood corresponds to x neighborhood.

Homeomorphism

A bicontinuous fn: \(X \to Y\). Respects topological properties.

Knots

A circular piece of thread. The simple ring or the unknot. The trefoil. Sketching knots. Strands: segments involved in a cross-over.

The 3 Reidemeister moves

See wikipedia article for figures. Sufficient and necessary to produce any valid deformation possible from a starting configuration.

Knot invariants

Property invariant to the Reidemesiter moves. 3 colorability of strands: Can assign 3 colors to strands such that all 3 colors are used; at each crossing, 3 or 1 colors seen.