GATE 2026 METRIC | SPACE | TOPOLOGY Live Question Solving скачать в хорошем качестве

GATE 2026 METRIC | SPACE | TOPOLOGY Live Question Solving

Definition of a Metric: The video explains the four essential properties of a metric ($d(x,y)$): Non-negativity: $d(x,y) \ge 0$ (1:40). Definiteness: $d(x,y) = 0$ if and only if $x=y$ (1:55). Symmetry: $d(x,y) = d(y,x)$ (2:23). Triangle Inequality: $d(x,y) \le d(x,z) + d(z,y)$ (2:57). Metric Space Examples: The video analyzes specific maps to determine if they constitute a metric space (6:04). Minimum Metric: $d(x,y) = \min(d_1(x,y), 3)$ is shown to satisfy the properties (6:46). Maximum Metric: $d(x,y) = \max(d_2(x,y), 3)$ fails the definiteness property because $x=y$ does not guarantee $d(x,y)=0$ (12:29). Fractional Metric: $d(x,y) = \frac{d_3(x,y)}{1+d_3(x,y)}$ (13:46). Topology Concepts (24:53 - 53:15) Definition of Topology: A collection of subsets ($ au$) of a set $X$ that satisfies three conditions: $X$ and $\emptyset$ are in $\tau$, arbitrary unions of sets in $\tau$ are in $\tau$, and finite intersections of sets in $\tau$ are in $\tau$ (25:07). Co-finite Topology: The video introduces the co-finite topology on $\mathbb{R}$, where a set is open if its complement is finite (28:47). Closed Sets: In the co-finite topology, finite subsets are closed sets (30:05). Separation Axioms: $T_0$ Space: Points can be separated by open sets (34:04). $T_1$ Space: Open sets exist to separate pairs of points uniquely (37:15). $T_2$ (Hausdorff) Space: Open sets are disjoint (38:59). The video proves that the co-finite topology on $\mathbb{R}$ is $T_1$ but not $T_2$ (42:04). Compactness and Connectedness: The video demonstrates that the co-finite topology on $\mathbb{R}$ is compact (45:26) and connected (50:22). Distinct Topologies Calculation (53:15 - End) Counting Topologies: The video concludes by calculating the number of distinct topologies on a 3-element set that consist of exactly four elements (53:15).