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 $\cup, \cap$ .

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 $U ∋ p$ . 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.